id	sid	tid	token	lemma	pos
ejpam-5887	1	1	european	european	PROPN
ejpam-5887	1	2	journal	journal	PROPN
ejpam-5887	1	3	of	of	ADP
ejpam-5887	1	4	pure	pure	ADJ
ejpam-5887	1	5	and	and	CCONJ
ejpam-5887	1	6	applied	applied	ADJ
ejpam-5887	1	7	mathematics	mathematic	NOUN
ejpam-5887	1	8	2025	2025	NUM
ejpam-5887	1	9	,	,	PUNCT
ejpam-5887	1	10	vol	vol	NOUN
ejpam-5887	1	11	.	.	PROPN
ejpam-5887	1	12	18	18	NUM
ejpam-5887	1	13	,	,	PUNCT
ejpam-5887	1	14	issue	issue	NOUN
ejpam-5887	1	15	2	2	NUM
ejpam-5887	1	16	,	,	PUNCT
ejpam-5887	1	17	article	article	NOUN
ejpam-5887	1	18	number	number	NOUN
ejpam-5887	1	19	5887	5887	NUM
ejpam-5887	1	20	issn	issn	VERB
ejpam-5887	1	21	1307	1307	NUM
ejpam-5887	1	22	-	-	SYM
ejpam-5887	1	23	5543	5543	NUM
ejpam-5887	1	24	–	–	PUNCT
ejpam-5887	1	25	ejpam.com	ejpam.com	X
ejpam-5887	1	26	published	publish	VERB
ejpam-5887	1	27	by	by	ADP
ejpam-5887	1	28	new	new	PROPN
ejpam-5887	1	29	york	york	PROPN
ejpam-5887	1	30	business	business	PROPN
ejpam-5887	1	31	global	global	PROPN
ejpam-5887	1	32	edge	edge	NOUN
ejpam-5887	1	33	k	k	NOUN
ejpam-5887	1	34	-	-	PUNCT
ejpam-5887	1	35	product	product	NOUN
ejpam-5887	1	36	cordial	cordial	ADJ
ejpam-5887	1	37	labeling	labeling	NOUN
ejpam-5887	1	38	of	of	ADP
ejpam-5887	1	39	graphs	graph	NOUN
ejpam-5887	1	40	n.	n.	PROPN
ejpam-5887	1	41	m.	m.	PROPN
ejpam-5887	1	42	noureldeen1,6	noureldeen1,6	PROPN
ejpam-5887	1	43	,	,	PUNCT
ejpam-5887	1	44	j.	j.	PROPN
ejpam-5887	1	45	jenisha2	jenisha2	PROPN
ejpam-5887	1	46	,	,	PUNCT
ejpam-5887	1	47	k.	k.	PROPN
ejpam-5887	1	48	jeya	jeya	PROPN
ejpam-5887	1	49	daisy3	daisy3	PROPN
ejpam-5887	1	50	,	,	PUNCT
ejpam-5887	1	51	p.	p.	PROPN
ejpam-5887	1	52	jeyanthi4,∗	jeyanthi4,∗	PROPN
ejpam-5887	1	53	,	,	PUNCT
ejpam-5887	1	54	m.	m.	PROPN
ejpam-5887	1	55	e.	e.	PROPN
ejpam-5887	1	56	abdel	abdel	PROPN
ejpam-5887	1	57	-	-	PUNCT
ejpam-5887	1	58	aal5	aal5	PROPN
ejpam-5887	1	59	1	1	NUM
ejpam-5887	1	60	department	department	NOUN
ejpam-5887	1	61	of	of	ADP
ejpam-5887	1	62	mathematics	mathematic	NOUN
ejpam-5887	1	63	,	,	PUNCT
ejpam-5887	1	64	college	college	NOUN
ejpam-5887	1	65	of	of	ADP
ejpam-5887	1	66	science	science	PROPN
ejpam-5887	1	67	,	,	PUNCT
ejpam-5887	1	68	taibah	taibah	PROPN
ejpam-5887	1	69	university	university	PROPN
ejpam-5887	1	70	,	,	PUNCT
ejpam-5887	1	71	madinah	madinah	PROPN
ejpam-5887	1	72	,	,	PUNCT
ejpam-5887	1	73	kingdom	kingdom	NOUN
ejpam-5887	1	74	of	of	ADP
ejpam-5887	1	75	saudi	saudi	PROPN
ejpam-5887	1	76	arabia	arabia	PROPN
ejpam-5887	1	77	2	2	NUM
ejpam-5887	1	78	research	research	NOUN
ejpam-5887	1	79	scholar	scholar	NOUN
ejpam-5887	1	80	(	(	PUNCT
ejpam-5887	1	81	reg.no	reg.no	PROPN
ejpam-5887	1	82	.	.	PUNCT
ejpam-5887	1	83	:	:	PUNCT
ejpam-5887	1	84	23213042092003	23213042092003	NUM
ejpam-5887	1	85	)	)	PUNCT
ejpam-5887	1	86	,	,	PUNCT
ejpam-5887	1	87	holy	holy	PROPN
ejpam-5887	1	88	cross	cross	PROPN
ejpam-5887	1	89	college	college	PROPN
ejpam-5887	1	90	(	(	PUNCT
ejpam-5887	1	91	autonomous	autonomous	ADJ
ejpam-5887	1	92	)	)	PUNCT
ejpam-5887	1	93	,	,	PUNCT
ejpam-5887	1	94	nagercoil	nagercoil	NOUN
ejpam-5887	1	95	629004	629004	NUM
ejpam-5887	1	96	,	,	PUNCT
ejpam-5887	1	97	tamilnadu	tamilnadu	NOUN
ejpam-5887	1	98	,	,	PUNCT
ejpam-5887	1	99	india	india	PROPN
ejpam-5887	1	100	,	,	PUNCT
ejpam-5887	1	101	affiliated	affiliate	VERB
ejpam-5887	1	102	to	to	ADP
ejpam-5887	1	103	manonmaniam	manonmaniam	PROPN
ejpam-5887	1	104	sundaranar	sundaranar	PROPN
ejpam-5887	1	105	university	university	PROPN
ejpam-5887	1	106	,	,	PUNCT
ejpam-5887	1	107	tirunelveli	tirunelveli	PROPN
ejpam-5887	1	108	627012	627012	NUM
ejpam-5887	1	109	,	,	PUNCT
ejpam-5887	1	110	tamilnadu	tamilnadu	NOUN
ejpam-5887	1	111	,	,	PUNCT
ejpam-5887	1	112	india	india	PROPN
ejpam-5887	1	113	3	3	NUM
ejpam-5887	1	114	pg	pg	NOUN
ejpam-5887	1	115	and	and	CCONJ
ejpam-5887	1	116	research	research	PROPN
ejpam-5887	1	117	department	department	PROPN
ejpam-5887	1	118	of	of	ADP
ejpam-5887	1	119	mathematics	mathematics	PROPN
ejpam-5887	1	120	,	,	PUNCT
ejpam-5887	1	121	holy	holy	PROPN
ejpam-5887	1	122	cross	cross	PROPN
ejpam-5887	1	123	college	college	PROPN
ejpam-5887	1	124	(	(	PUNCT
ejpam-5887	1	125	autonomous	autonomous	ADJ
ejpam-5887	1	126	)	)	PUNCT
ejpam-5887	1	127	,	,	PUNCT
ejpam-5887	1	128	nagercoil	nagercoil	NOUN
ejpam-5887	1	129	629004	629004	NUM
ejpam-5887	1	130	,	,	PUNCT
ejpam-5887	1	131	tamilnadu	tamilnadu	NOUN
ejpam-5887	1	132	,	,	PUNCT
ejpam-5887	1	133	india	india	PROPN
ejpam-5887	1	134	4	4	NUM
ejpam-5887	1	135	research	research	NOUN
ejpam-5887	1	136	centre	centre	NOUN
ejpam-5887	1	137	,	,	PUNCT
ejpam-5887	1	138	department	department	NOUN
ejpam-5887	1	139	of	of	ADP
ejpam-5887	1	140	mathematics	mathematic	NOUN
ejpam-5887	1	141	,	,	PUNCT
ejpam-5887	1	142	govindammal	govindammal	ADJ
ejpam-5887	1	143	aditanar	aditanar	ADJ
ejpam-5887	1	144	college	college	NOUN
ejpam-5887	1	145	for	for	ADP
ejpam-5887	1	146	women	woman	NOUN
ejpam-5887	1	147	,	,	PUNCT
ejpam-5887	1	148	tiruchendur	tiruchendur	PROPN
ejpam-5887	1	149	628215	628215	NUM
ejpam-5887	1	150	,	,	PUNCT
ejpam-5887	1	151	tamilnadu	tamilnadu	NOUN
ejpam-5887	1	152	,	,	PUNCT
ejpam-5887	1	153	india	india	PROPN
ejpam-5887	1	154	.	.	PROPN
ejpam-5887	2	1	5	5	NUM
ejpam-5887	2	2	department	department	NOUN
ejpam-5887	2	3	of	of	ADP
ejpam-5887	2	4	mathematics	mathematic	NOUN
ejpam-5887	2	5	,	,	PUNCT
ejpam-5887	2	6	faculty	faculty	NOUN
ejpam-5887	2	7	of	of	ADP
ejpam-5887	2	8	science	science	NOUN
ejpam-5887	2	9	,	,	PUNCT
ejpam-5887	2	10	benha	benha	VERB
ejpam-5887	2	11	university	university	NOUN
ejpam-5887	2	12	,	,	PUNCT
ejpam-5887	2	13	benha	benha	VERB
ejpam-5887	2	14	13518	13518	NUM
ejpam-5887	2	15	,	,	PUNCT
ejpam-5887	2	16	egypt	egypt	PROPN
ejpam-5887	2	17	6	6	NUM
ejpam-5887	2	18	department	department	NOUN
ejpam-5887	2	19	of	of	ADP
ejpam-5887	2	20	mathematics	mathematic	NOUN
ejpam-5887	2	21	,	,	PUNCT
ejpam-5887	2	22	women	woman	NOUN
ejpam-5887	2	23	’s	’s	PART
ejpam-5887	2	24	college	college	PROPN
ejpam-5887	2	25	of	of	ADP
ejpam-5887	2	26	arts	art	NOUN
ejpam-5887	2	27	,	,	PUNCT
ejpam-5887	2	28	sciences	science	NOUN
ejpam-5887	2	29	and	and	CCONJ
ejpam-5887	2	30	education	education	NOUN
ejpam-5887	2	31	,	,	PUNCT
ejpam-5887	2	32	ain	ain	PROPN
ejpam-5887	2	33	shams	shams	PROPN
ejpam-5887	2	34	university	university	PROPN
ejpam-5887	2	35	,	,	PUNCT
ejpam-5887	2	36	egypt	egypt	PROPN
ejpam-5887	2	37	abstract	abstract	PROPN
ejpam-5887	2	38	.	.	PUNCT
ejpam-5887	3	1	in	in	ADP
ejpam-5887	3	2	this	this	DET
ejpam-5887	3	3	paper	paper	NOUN
ejpam-5887	3	4	,	,	PUNCT
ejpam-5887	3	5	we	we	PRON
ejpam-5887	3	6	introduce	introduce	VERB
ejpam-5887	3	7	a	a	DET
ejpam-5887	3	8	new	new	ADJ
ejpam-5887	3	9	labeling	labeling	NOUN
ejpam-5887	3	10	namely	namely	ADV
ejpam-5887	3	11	‘	'	PUNCT
ejpam-5887	3	12	edge	edge	ADJ
ejpam-5887	3	13	k	k	ADJ
ejpam-5887	3	14	-	-	PUNCT
ejpam-5887	3	15	product	product	NOUN
ejpam-5887	3	16	cordial	cordial	ADJ
ejpam-5887	3	17	labeling	labeling	NOUN
ejpam-5887	3	18	’	'	PUNCT
ejpam-5887	3	19	as	as	SCONJ
ejpam-5887	3	20	follows	follow	VERB
ejpam-5887	3	21	:	:	PUNCT
ejpam-5887	3	22	for	for	ADP
ejpam-5887	3	23	a	a	DET
ejpam-5887	3	24	graph	graph	NOUN
ejpam-5887	3	25	g	g	NOUN
ejpam-5887	3	26	=	=	PUNCT
ejpam-5887	3	27	(	(	PUNCT
ejpam-5887	3	28	v	v	NOUN
ejpam-5887	3	29	(	(	PUNCT
ejpam-5887	3	30	g	g	NOUN
ejpam-5887	3	31	)	)	PUNCT
ejpam-5887	3	32	,	,	PUNCT
ejpam-5887	3	33	e(g	e(g	PROPN
ejpam-5887	3	34	)	)	PUNCT
ejpam-5887	3	35	)	)	PUNCT
ejpam-5887	4	1	having	have	VERB
ejpam-5887	4	2	no	no	DET
ejpam-5887	4	3	isolated	isolated	ADJ
ejpam-5887	4	4	vertex	vertex	NOUN
ejpam-5887	4	5	,	,	PUNCT
ejpam-5887	4	6	an	an	DET
ejpam-5887	4	7	edge	edge	NOUN
ejpam-5887	4	8	labeling	labeling	NOUN
ejpam-5887	5	1	f	f	NOUN
ejpam-5887	5	2	:	:	PUNCT
ejpam-5887	5	3	e(g	e(g	PROPN
ejpam-5887	5	4	)	)	PUNCT
ejpam-5887	5	5	→	→	PUNCT
ejpam-5887	5	6	{	{	PUNCT
ejpam-5887	5	7	0	0	NUM
ejpam-5887	5	8	,	,	PUNCT
ejpam-5887	5	9	1	1	NUM
ejpam-5887	5	10	,	,	PUNCT
ejpam-5887	5	11	...	...	PUNCT
ejpam-5887	5	12	,	,	PUNCT
ejpam-5887	5	13	k	k	PROPN
ejpam-5887	6	1	−	−	PROPN
ejpam-5887	6	2	1	1	NUM
ejpam-5887	6	3	}	}	PUNCT
ejpam-5887	6	4	,	,	PUNCT
ejpam-5887	6	5	where	where	SCONJ
ejpam-5887	6	6	k	k	PROPN
ejpam-5887	6	7	>	>	X
ejpam-5887	6	8	1	1	NUM
ejpam-5887	6	9	is	be	AUX
ejpam-5887	6	10	an	an	DET
ejpam-5887	6	11	integer	integer	NOUN
ejpam-5887	6	12	,	,	PUNCT
ejpam-5887	6	13	is	be	AUX
ejpam-5887	6	14	said	say	VERB
ejpam-5887	6	15	to	to	PART
ejpam-5887	6	16	be	be	AUX
ejpam-5887	6	17	an	an	DET
ejpam-5887	6	18	edge	edge	NOUN
ejpam-5887	6	19	k	k	NOUN
ejpam-5887	6	20	-	-	PUNCT
ejpam-5887	6	21	product	product	NOUN
ejpam-5887	6	22	cordial	cordial	ADJ
ejpam-5887	6	23	labeling	labeling	NOUN
ejpam-5887	6	24	if	if	SCONJ
ejpam-5887	6	25	it	it	PRON
ejpam-5887	6	26	induces	induce	VERB
ejpam-5887	6	27	a	a	DET
ejpam-5887	6	28	vertex	vertex	NOUN
ejpam-5887	6	29	labeling	labeling	NOUN
ejpam-5887	6	30	f⋆	f⋆	NUM
ejpam-5887	6	31	:	:	PUNCT
ejpam-5887	6	32	v	v	NOUN
ejpam-5887	6	33	(	(	PUNCT
ejpam-5887	6	34	g	g	NOUN
ejpam-5887	6	35	)	)	PUNCT
ejpam-5887	6	36	→	→	SYM
ejpam-5887	6	37	{	{	PUNCT
ejpam-5887	6	38	0	0	NUM
ejpam-5887	6	39	,	,	PUNCT
ejpam-5887	6	40	1	1	NUM
ejpam-5887	6	41	,	,	PUNCT
ejpam-5887	6	42	...	...	PUNCT
ejpam-5887	6	43	,	,	PUNCT
ejpam-5887	7	1	k	k	PROPN
ejpam-5887	7	2	−	−	PROPN
ejpam-5887	7	3	1	1	NUM
ejpam-5887	7	4	}	}	PUNCT
ejpam-5887	7	5	defined	define	VERB
ejpam-5887	7	6	by	by	ADP
ejpam-5887	7	7	f⋆(v	f⋆(v	NOUN
ejpam-5887	7	8	)	)	PUNCT
ejpam-5887	7	9	=	=	SYM
ejpam-5887	7	10	∏	∏	PROPN
ejpam-5887	7	11	uv∈e(g	uv∈e(g	NOUN
ejpam-5887	7	12	)	)	PUNCT
ejpam-5887	7	13	f(uv)(mod	f(uv)(mod	ADP
ejpam-5887	7	14	k	k	X
ejpam-5887	7	15	)	)	PUNCT
ejpam-5887	7	16	satisfies	satisfie	NOUN
ejpam-5887	7	17	|ef	|ef	NUM
ejpam-5887	7	18	(	(	PUNCT
ejpam-5887	7	19	i)−	i)−	PROPN
ejpam-5887	7	20	ef	ef	PROPN
ejpam-5887	7	21	(	(	PUNCT
ejpam-5887	7	22	j)|	j)|	NOUN
ejpam-5887	7	23	≤	≤	NUM
ejpam-5887	7	24	1	1	NUM
ejpam-5887	7	25	and	and	CCONJ
ejpam-5887	7	26	|vf⋆(i)−	|vf⋆(i)−	NOUN
ejpam-5887	7	27	vf⋆(j)|	vf⋆(j)|	NOUN
ejpam-5887	7	28	≤	≤	NUM
ejpam-5887	7	29	1	1	NUM
ejpam-5887	7	30	for	for	ADP
ejpam-5887	7	31	i	i	PRON
ejpam-5887	7	32	,	,	PUNCT
ejpam-5887	7	33	j	j	PROPN
ejpam-5887	7	34	∈	∈	PROPN
ejpam-5887	7	35	{	{	PUNCT
ejpam-5887	7	36	0	0	NUM
ejpam-5887	7	37	,	,	PUNCT
ejpam-5887	7	38	1	1	NUM
ejpam-5887	7	39	,	,	PUNCT
ejpam-5887	7	40	...	...	PUNCT
ejpam-5887	7	41	,	,	PUNCT
ejpam-5887	7	42	k	k	PROPN
ejpam-5887	7	43	−	−	PROPN
ejpam-5887	7	44	1	1	NUM
ejpam-5887	7	45	}	}	PUNCT
ejpam-5887	7	46	,	,	PUNCT
ejpam-5887	7	47	where	where	SCONJ
ejpam-5887	7	48	ef	ef	PROPN
ejpam-5887	7	49	(	(	PUNCT
ejpam-5887	7	50	i	i	PROPN
ejpam-5887	7	51	)	)	PUNCT
ejpam-5887	7	52	and	and	CCONJ
ejpam-5887	7	53	vf⋆(i	vf⋆(i	PROPN
ejpam-5887	7	54	)	)	PUNCT
ejpam-5887	7	55	denote	denote	VERB
ejpam-5887	7	56	the	the	DET
ejpam-5887	7	57	number	number	NOUN
ejpam-5887	7	58	of	of	ADP
ejpam-5887	7	59	edges	edge	NOUN
ejpam-5887	7	60	and	and	CCONJ
ejpam-5887	7	61	vertices	vertex	NOUN
ejpam-5887	7	62	respectively	respectively	ADV
ejpam-5887	7	63	having	have	VERB
ejpam-5887	7	64	a	a	DET
ejpam-5887	7	65	label	label	NOUN
ejpam-5887	8	1	i	i	PRON
ejpam-5887	8	2	(	(	PUNCT
ejpam-5887	8	3	i	i	NOUN
ejpam-5887	8	4	=	=	NOUN
ejpam-5887	8	5	0	0	NUM
ejpam-5887	8	6	,	,	PUNCT
ejpam-5887	8	7	1	1	NUM
ejpam-5887	8	8	,	,	PUNCT
ejpam-5887	8	9	...	...	PUNCT
ejpam-5887	8	10	,	,	PUNCT
ejpam-5887	8	11	k	k	PROPN
ejpam-5887	8	12	−	−	PROPN
ejpam-5887	8	13	1	1	NUM
ejpam-5887	8	14	)	)	PUNCT
ejpam-5887	8	15	.	.	PUNCT
ejpam-5887	9	1	further	far	ADV
ejpam-5887	9	2	,	,	PUNCT
ejpam-5887	9	3	we	we	PRON
ejpam-5887	9	4	study	study	VERB
ejpam-5887	9	5	the	the	DET
ejpam-5887	9	6	edge	edge	NOUN
ejpam-5887	9	7	k	k	NOUN
ejpam-5887	9	8	-	-	PUNCT
ejpam-5887	9	9	product	product	NOUN
ejpam-5887	9	10	cordial	cordial	ADJ
ejpam-5887	9	11	behavior	behavior	NOUN
ejpam-5887	9	12	of	of	ADP
ejpam-5887	9	13	star	star	PROPN
ejpam-5887	9	14	,	,	PUNCT
ejpam-5887	9	15	bistar	bistar	PROPN
ejpam-5887	9	16	,	,	PUNCT
ejpam-5887	9	17	shadow	shadow	NOUN
ejpam-5887	9	18	and	and	CCONJ
ejpam-5887	9	19	splitting	splitting	NOUN
ejpam-5887	9	20	graph	graph	NOUN
ejpam-5887	9	21	of	of	ADP
ejpam-5887	9	22	star	star	NOUN
ejpam-5887	9	23	,	,	PUNCT
ejpam-5887	9	24	path	path	NOUN
ejpam-5887	9	25	union	union	PROPN
ejpam-5887	9	26	of	of	ADP
ejpam-5887	9	27	star	star	PROPN
ejpam-5887	9	28	,	,	PUNCT
ejpam-5887	9	29	bistar	bistar	NOUN
ejpam-5887	9	30	and	and	CCONJ
ejpam-5887	9	31	cycle	cycle	NOUN
ejpam-5887	9	32	graphs	graph	NOUN
ejpam-5887	9	33	.	.	PUNCT
ejpam-5887	10	1	2020	2020	NUM
ejpam-5887	10	2	mathematics	mathematic	NOUN
ejpam-5887	10	3	subject	subject	NOUN
ejpam-5887	10	4	classifications	classification	NOUN
ejpam-5887	10	5	:	:	PUNCT
ejpam-5887	10	6	05c78	05c78	NUM
ejpam-5887	10	7	key	key	ADJ
ejpam-5887	10	8	words	word	NOUN
ejpam-5887	10	9	and	and	CCONJ
ejpam-5887	10	10	phrases	phrase	NOUN
ejpam-5887	10	11	:	:	PUNCT
ejpam-5887	10	12	product	product	VERB
ejpam-5887	10	13	cordial	cordial	ADJ
ejpam-5887	10	14	labeling	labeling	NOUN
ejpam-5887	10	15	,	,	PUNCT
ejpam-5887	10	16	k	k	ADJ
ejpam-5887	10	17	-	-	PUNCT
ejpam-5887	10	18	product	product	NOUN
ejpam-5887	10	19	cordial	cordial	ADJ
ejpam-5887	10	20	labeling	labeling	NOUN
ejpam-5887	10	21	,	,	PUNCT
ejpam-5887	10	22	edge	edge	NOUN
ejpam-5887	10	23	k	k	ADJ
ejpam-5887	10	24	-	-	PUNCT
ejpam-5887	10	25	product	product	NOUN
ejpam-5887	10	26	cordial	cordial	ADJ
ejpam-5887	10	27	labeling	labeling	NOUN
ejpam-5887	10	28	,	,	PUNCT
ejpam-5887	10	29	shadow	shadow	NOUN
ejpam-5887	10	30	graph	graph	NOUN
ejpam-5887	10	31	,	,	PUNCT
ejpam-5887	10	32	splitting	splitting	NOUN
ejpam-5887	10	33	graph	graph	NOUN
ejpam-5887	10	34	,	,	PUNCT
ejpam-5887	10	35	path	path	NOUN
ejpam-5887	10	36	union	union	NOUN
ejpam-5887	10	37	of	of	ADP
ejpam-5887	10	38	graph	graph	NOUN
ejpam-5887	10	39	1	1	NUM
ejpam-5887	10	40	.	.	PUNCT
ejpam-5887	10	41	introduction	introduction	NOUN
ejpam-5887	10	42	in	in	ADP
ejpam-5887	10	43	mathematics	mathematic	NOUN
ejpam-5887	10	44	,	,	PUNCT
ejpam-5887	10	45	the	the	DET
ejpam-5887	10	46	field	field	NOUN
ejpam-5887	10	47	of	of	ADP
ejpam-5887	10	48	graph	graph	NOUN
ejpam-5887	10	49	theory	theory	NOUN
ejpam-5887	10	50	revolves	revolve	VERB
ejpam-5887	10	51	around	around	ADP
ejpam-5887	10	52	the	the	DET
ejpam-5887	10	53	examination	examination	NOUN
ejpam-5887	10	54	of	of	ADP
ejpam-5887	10	55	graphs	graph	NOUN
ejpam-5887	10	56	,	,	PUNCT
ejpam-5887	10	57	which	which	PRON
ejpam-5887	10	58	are	be	AUX
ejpam-5887	10	59	the	the	DET
ejpam-5887	10	60	fundamental	fundamental	ADJ
ejpam-5887	10	61	objects	object	NOUN
ejpam-5887	10	62	within	within	ADP
ejpam-5887	10	63	discrete	discrete	ADJ
ejpam-5887	10	64	mathematics	mathematic	NOUN
ejpam-5887	10	65	.	.	PUNCT
ejpam-5887	11	1	over	over	ADP
ejpam-5887	11	2	the	the	DET
ejpam-5887	11	3	past	past	ADJ
ejpam-5887	11	4	six	six	NUM
ejpam-5887	11	5	decades	decade	NOUN
ejpam-5887	11	6	,	,	PUNCT
ejpam-5887	11	7	∗corresponding	∗corresponde	VERB
ejpam-5887	11	8	author	author	NOUN
ejpam-5887	11	9	.	.	PUNCT
ejpam-5887	12	1	doi	doi	NOUN
ejpam-5887	12	2	:	:	PUNCT
ejpam-5887	12	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5887	https://doi.org/10.29020/nybg.ejpam.v18i2.5887	ADJ
ejpam-5887	12	4	email	email	NOUN
ejpam-5887	12	5	addresses	address	NOUN
ejpam-5887	12	6	:	:	PUNCT
ejpam-5887	12	7	neldeen@taibahu.edu.sa	neldeen@taibahu.edu.sa	PROPN
ejpam-5887	12	8	(	(	PUNCT
ejpam-5887	12	9	n.	n.	PROPN
ejpam-5887	12	10	m.	m.	PROPN
ejpam-5887	12	11	noureldeen	noureldeen	PROPN
ejpam-5887	12	12	)	)	PUNCT
ejpam-5887	12	13	,	,	PUNCT
ejpam-5887	12	14	jenishaelston@gmail.com	jenishaelston@gmail.com	PROPN
ejpam-5887	12	15	(	(	PUNCT
ejpam-5887	12	16	j.	j.	PROPN
ejpam-5887	12	17	jenisha	jenisha	PROPN
ejpam-5887	12	18	)	)	PUNCT
ejpam-5887	12	19	,	,	PUNCT
ejpam-5887	12	20	jeyadaisy@holycrossngl.edu.in	jeyadaisy@holycrossngl.edu.in	PROPN
ejpam-5887	12	21	(	(	PUNCT
ejpam-5887	12	22	k.	k.	NOUN
ejpam-5887	12	23	jeya	jeya	PROPN
ejpam-5887	12	24	daisy	daisy	PROPN
ejpam-5887	12	25	)	)	PUNCT
ejpam-5887	12	26	,	,	PUNCT
ejpam-5887	12	27	jeyajeyanthi@rediffmail.com	jeyajeyanthi@rediffmail.com	X
ejpam-5887	13	1	(	(	PUNCT
ejpam-5887	13	2	p.	p.	NOUN
ejpam-5887	13	3	jeyanthi	jeyanthi	PROPN
ejpam-5887	13	4	)	)	PUNCT
ejpam-5887	13	5	,	,	PUNCT
ejpam-5887	13	6	mohamed.abdelghani@fsc.bu.edu.eg	mohamed.abdelghani@fsc.bu.edu.eg	PROPN
ejpam-5887	13	7	(	(	PUNCT
ejpam-5887	13	8	m.	m.	PROPN
ejpam-5887	13	9	e.	e.	PROPN
ejpam-5887	13	10	abdel	abdel	PROPN
ejpam-5887	13	11	-	-	PUNCT
ejpam-5887	13	12	aal	aal	PROPN
ejpam-5887	13	13	)	)	PUNCT
ejpam-5887	13	14	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5887	14	1	1	1	NUM
ejpam-5887	15	1	copyright	copyright	NOUN
ejpam-5887	15	2	:	:	PUNCT
ejpam-5887	15	3	©	©	PROPN
ejpam-5887	15	4	2025	2025	NUM
ejpam-5887	15	5	the	the	DET
ejpam-5887	15	6	author(s	author(s	NOUN
ejpam-5887	15	7	)	)	PUNCT
ejpam-5887	15	8	.	.	PUNCT
ejpam-5887	16	1	(	(	PUNCT
ejpam-5887	16	2	cc	cc	NOUN
ejpam-5887	16	3	by	by	ADP
ejpam-5887	16	4	-	-	PUNCT
ejpam-5887	16	5	nc	nc	PROPN
ejpam-5887	16	6	4.0	4.0	NUM
ejpam-5887	16	7	)	)	PUNCT
ejpam-5887	16	8	n.	n.	NOUN
ejpam-5887	16	9	m.	m.	NOUN
ejpam-5887	16	10	noureldeen	noureldeen	NOUN
ejpam-5887	16	11	et	et	PROPN
ejpam-5887	16	12	al	al	PROPN
ejpam-5887	16	13	.	.	PUNCT
ejpam-5887	16	14	/	/	SYM
ejpam-5887	16	15	eur	eur	PROPN
ejpam-5887	16	16	.	.	PUNCT
ejpam-5887	17	1	j.	j.	PROPN
ejpam-5887	17	2	pure	pure	PROPN
ejpam-5887	17	3	appl	appl	PROPN
ejpam-5887	17	4	.	.	PROPN
ejpam-5887	17	5	math	math	PROPN
ejpam-5887	17	6	,	,	PUNCT
ejpam-5887	17	7	18	18	NUM
ejpam-5887	17	8	(	(	PUNCT
ejpam-5887	17	9	2	2	NUM
ejpam-5887	17	10	)	)	PUNCT
ejpam-5887	17	11	(	(	PUNCT
ejpam-5887	17	12	2025	2025	NUM
ejpam-5887	17	13	)	)	PUNCT
ejpam-5887	17	14	,	,	PUNCT
ejpam-5887	17	15	5887	5887	NUM
ejpam-5887	17	16	2	2	NUM
ejpam-5887	17	17	of	of	ADP
ejpam-5887	17	18	21	21	NUM
ejpam-5887	17	19	one	one	NUM
ejpam-5887	17	20	particular	particular	ADJ
ejpam-5887	17	21	aspect	aspect	NOUN
ejpam-5887	17	22	of	of	ADP
ejpam-5887	17	23	graph	graph	NOUN
ejpam-5887	17	24	theory	theory	NOUN
ejpam-5887	17	25	called	call	VERB
ejpam-5887	17	26	graph	graph	NOUN
ejpam-5887	17	27	labeling	labeling	NOUN
ejpam-5887	17	28	,	,	PUNCT
ejpam-5887	17	29	has	have	AUX
ejpam-5887	17	30	gained	gain	VERB
ejpam-5887	17	31	significant	significant	ADJ
ejpam-5887	17	32	popularity	popularity	NOUN
ejpam-5887	17	33	due	due	ADP
ejpam-5887	17	34	to	to	ADP
ejpam-5887	17	35	its	its	PRON
ejpam-5887	17	36	diverse	diverse	ADJ
ejpam-5887	17	37	applications	application	NOUN
ejpam-5887	17	38	.	.	PUNCT
ejpam-5887	18	1	labeling	labeling	NOUN
ejpam-5887	18	2	involves	involve	VERB
ejpam-5887	18	3	assigning	assign	VERB
ejpam-5887	18	4	real	real	ADJ
ejpam-5887	18	5	numbers	number	NOUN
ejpam-5887	18	6	,	,	PUNCT
ejpam-5887	18	7	typically	typically	ADV
ejpam-5887	18	8	positive	positive	ADJ
ejpam-5887	18	9	integers	integer	NOUN
ejpam-5887	18	10	,	,	PUNCT
ejpam-5887	18	11	to	to	ADP
ejpam-5887	18	12	the	the	DET
ejpam-5887	18	13	elements	element	NOUN
ejpam-5887	18	14	of	of	ADP
ejpam-5887	18	15	a	a	DET
ejpam-5887	18	16	graph	graph	NOUN
ejpam-5887	18	17	.	.	PUNCT
ejpam-5887	19	1	in	in	ADP
ejpam-5887	19	2	1967	1967	NUM
ejpam-5887	19	3	,	,	PUNCT
ejpam-5887	19	4	rosa	rosa	PROPN
ejpam-5887	19	5	[	[	X
ejpam-5887	19	6	10	10	NUM
ejpam-5887	19	7	]	]	PUNCT
ejpam-5887	19	8	published	publish	VERB
ejpam-5887	19	9	an	an	DET
ejpam-5887	19	10	influential	influential	ADJ
ejpam-5887	19	11	paper	paper	NOUN
ejpam-5887	19	12	that	that	PRON
ejpam-5887	19	13	laid	lay	VERB
ejpam-5887	19	14	the	the	DET
ejpam-5887	19	15	groundwork	groundwork	NOUN
ejpam-5887	19	16	for	for	ADP
ejpam-5887	19	17	various	various	ADJ
ejpam-5887	19	18	graph	graph	NOUN
ejpam-5887	19	19	labeling	labeling	NOUN
ejpam-5887	19	20	problems	problem	NOUN
ejpam-5887	19	21	.	.	PUNCT
ejpam-5887	20	1	since	since	SCONJ
ejpam-5887	20	2	then	then	ADV
ejpam-5887	20	3	,	,	PUNCT
ejpam-5887	20	4	numerous	numerous	ADJ
ejpam-5887	20	5	authors	author	NOUN
ejpam-5887	20	6	have	have	AUX
ejpam-5887	20	7	delved	delve	VERB
ejpam-5887	20	8	into	into	ADP
ejpam-5887	20	9	researching	research	VERB
ejpam-5887	20	10	various	various	ADJ
ejpam-5887	20	11	graph	graph	NOUN
ejpam-5887	20	12	labeling	labeling	NOUN
ejpam-5887	20	13	techniques	technique	NOUN
ejpam-5887	20	14	and	and	CCONJ
ejpam-5887	20	15	a	a	DET
ejpam-5887	20	16	detailed	detailed	ADJ
ejpam-5887	20	17	survey	survey	NOUN
ejpam-5887	20	18	is	be	AUX
ejpam-5887	20	19	available	available	ADJ
ejpam-5887	20	20	in	in	ADP
ejpam-5887	20	21	[	[	X
ejpam-5887	20	22	4	4	NUM
ejpam-5887	20	23	]	]	PUNCT
ejpam-5887	20	24	.	.	PUNCT
ejpam-5887	21	1	among	among	ADP
ejpam-5887	21	2	these	these	DET
ejpam-5887	21	3	labeling	labeling	NOUN
ejpam-5887	21	4	techniques	technique	NOUN
ejpam-5887	21	5	,	,	PUNCT
ejpam-5887	21	6	‘	'	PUNCT
ejpam-5887	21	7	cordial	cordial	ADJ
ejpam-5887	21	8	labeling	labeling	NOUN
ejpam-5887	21	9	’	'	PUNCT
ejpam-5887	21	10	by	by	ADP
ejpam-5887	21	11	cahit	cahit	ADJ
ejpam-5887	21	12	[	[	X
ejpam-5887	21	13	3	3	NUM
ejpam-5887	21	14	]	]	PUNCT
ejpam-5887	21	15	stands	stand	VERB
ejpam-5887	21	16	out	out	ADP
ejpam-5887	21	17	as	as	ADP
ejpam-5887	21	18	a	a	DET
ejpam-5887	21	19	less	less	ADV
ejpam-5887	21	20	stringent	stringent	ADJ
ejpam-5887	21	21	version	version	NOUN
ejpam-5887	21	22	compared	compare	VERB
ejpam-5887	21	23	to	to	ADP
ejpam-5887	21	24	graceful	graceful	ADJ
ejpam-5887	21	25	and	and	CCONJ
ejpam-5887	21	26	harmonious	harmonious	ADJ
ejpam-5887	21	27	labeling	labeling	NOUN
ejpam-5887	21	28	.	.	PUNCT
ejpam-5887	22	1	of	of	ADP
ejpam-5887	22	2	these	these	PRON
ejpam-5887	22	3	,	,	PUNCT
ejpam-5887	22	4	graceful	graceful	ADJ
ejpam-5887	22	5	labeling	labeling	NOUN
ejpam-5887	22	6	is	be	AUX
ejpam-5887	22	7	more	more	ADV
ejpam-5887	22	8	popular	popular	ADJ
ejpam-5887	22	9	since	since	SCONJ
ejpam-5887	22	10	it	it	PRON
ejpam-5887	22	11	has	have	VERB
ejpam-5887	22	12	various	various	ADJ
ejpam-5887	22	13	practical	practical	ADJ
ejpam-5887	22	14	applications	application	NOUN
ejpam-5887	22	15	.	.	PUNCT
ejpam-5887	23	1	cordial	cordial	ADJ
ejpam-5887	23	2	labeling	labeling	NOUN
ejpam-5887	23	3	has	have	VERB
ejpam-5887	23	4	potential	potential	ADJ
ejpam-5887	23	5	applications	application	NOUN
ejpam-5887	23	6	in	in	ADP
ejpam-5887	23	7	areas	area	NOUN
ejpam-5887	23	8	such	such	ADJ
ejpam-5887	23	9	as	as	ADP
ejpam-5887	23	10	network	network	NOUN
ejpam-5887	23	11	design	design	NOUN
ejpam-5887	23	12	,	,	PUNCT
ejpam-5887	23	13	error	error	NOUN
ejpam-5887	23	14	-	-	PUNCT
ejpam-5887	23	15	correcting	correct	VERB
ejpam-5887	23	16	codes	code	NOUN
ejpam-5887	23	17	,	,	PUNCT
ejpam-5887	23	18	and	and	CCONJ
ejpam-5887	23	19	cryptography	cryptography	NOUN
ejpam-5887	23	20	.	.	PUNCT
ejpam-5887	24	1	in	in	ADP
ejpam-5887	24	2	particular	particular	ADJ
ejpam-5887	24	3	,	,	PUNCT
ejpam-5887	24	4	cordial	cordial	ADJ
ejpam-5887	24	5	labeling	labeling	NOUN
ejpam-5887	24	6	could	could	AUX
ejpam-5887	24	7	be	be	AUX
ejpam-5887	24	8	useful	useful	ADJ
ejpam-5887	24	9	in	in	ADP
ejpam-5887	24	10	designing	design	VERB
ejpam-5887	24	11	efficient	efficient	ADJ
ejpam-5887	24	12	communication	communication	NOUN
ejpam-5887	24	13	protocols	protocol	NOUN
ejpam-5887	24	14	where	where	SCONJ
ejpam-5887	24	15	balancing	balance	VERB
ejpam-5887	24	16	two	two	NUM
ejpam-5887	24	17	types	type	NOUN
ejpam-5887	24	18	of	of	ADP
ejpam-5887	24	19	nodes	node	NOUN
ejpam-5887	24	20	(	(	PUNCT
ejpam-5887	24	21	positive	positive	ADJ
ejpam-5887	24	22	and	and	CCONJ
ejpam-5887	24	23	negative	negative	ADJ
ejpam-5887	24	24	)	)	PUNCT
ejpam-5887	24	25	is	be	AUX
ejpam-5887	24	26	essential	essential	ADJ
ejpam-5887	24	27	.	.	PUNCT
ejpam-5887	25	1	in	in	ADP
ejpam-5887	25	2	the	the	DET
ejpam-5887	25	3	subsequent	subsequent	ADJ
ejpam-5887	25	4	years	year	NOUN
ejpam-5887	25	5	,	,	PUNCT
ejpam-5887	25	6	several	several	ADJ
ejpam-5887	25	7	variants	variant	NOUN
ejpam-5887	25	8	of	of	ADP
ejpam-5887	25	9	cordial	cordial	ADJ
ejpam-5887	25	10	labeling	labeling	NOUN
ejpam-5887	25	11	,	,	PUNCT
ejpam-5887	25	12	such	such	ADJ
ejpam-5887	25	13	as	as	ADP
ejpam-5887	25	14	,	,	PUNCT
ejpam-5887	25	15	‘	'	PUNCT
ejpam-5887	25	16	product	product	NOUN
ejpam-5887	25	17	cordial	cordial	ADJ
ejpam-5887	25	18	labeling	labeling	NOUN
ejpam-5887	25	19	’	'	PUNCT
ejpam-5887	25	20	,	,	PUNCT
ejpam-5887	25	21	‘	'	PUNCT
ejpam-5887	25	22	k	k	ADJ
ejpam-5887	25	23	-	-	ADJ
ejpam-5887	25	24	product	product	NOUN
ejpam-5887	25	25	cordial	cordial	ADJ
ejpam-5887	25	26	labeling	labeling	NOUN
ejpam-5887	25	27	’	'	PUNCT
ejpam-5887	25	28	,	,	PUNCT
ejpam-5887	25	29	‘	'	PUNCT
ejpam-5887	25	30	edge	edge	NOUN
ejpam-5887	25	31	product	product	NOUN
ejpam-5887	25	32	cordial	cordial	ADJ
ejpam-5887	25	33	labeling	labeling	NOUN
ejpam-5887	25	34	’	'	PUNCT
ejpam-5887	25	35	and	and	CCONJ
ejpam-5887	25	36	more	more	ADJ
ejpam-5887	25	37	are	be	AUX
ejpam-5887	25	38	introduced	introduce	VERB
ejpam-5887	25	39	.	.	PUNCT
ejpam-5887	26	1	in	in	ADP
ejpam-5887	26	2	‘	'	PUNCT
ejpam-5887	26	3	edge	edge	NOUN
ejpam-5887	26	4	product	product	NOUN
ejpam-5887	26	5	cordial	cordial	ADJ
ejpam-5887	26	6	labeling	labeling	NOUN
ejpam-5887	26	7	’	'	PUNCT
ejpam-5887	27	1	[	[	X
ejpam-5887	27	2	11	11	NUM
ejpam-5887	27	3	]	]	PUNCT
ejpam-5887	27	4	,	,	PUNCT
ejpam-5887	27	5	the	the	DET
ejpam-5887	27	6	roles	role	NOUN
ejpam-5887	27	7	of	of	ADP
ejpam-5887	27	8	vertices	vertex	NOUN
ejpam-5887	27	9	and	and	CCONJ
ejpam-5887	27	10	edges	edge	NOUN
ejpam-5887	27	11	in	in	ADP
ejpam-5887	27	12	product	product	NOUN
ejpam-5887	27	13	cordial	cordial	ADJ
ejpam-5887	27	14	labeling	labeling	NOUN
ejpam-5887	27	15	[	[	X
ejpam-5887	27	16	6	6	NUM
ejpam-5887	27	17	]	]	PUNCT
ejpam-5887	27	18	are	be	AUX
ejpam-5887	27	19	swapped	swap	VERB
ejpam-5887	27	20	.	.	PUNCT
ejpam-5887	28	1	building	build	VERB
ejpam-5887	28	2	on	on	ADP
ejpam-5887	28	3	this	this	DET
ejpam-5887	28	4	notion	notion	NOUN
ejpam-5887	28	5	,	,	PUNCT
ejpam-5887	28	6	several	several	ADJ
ejpam-5887	28	7	results	result	NOUN
ejpam-5887	28	8	have	have	AUX
ejpam-5887	28	9	been	be	AUX
ejpam-5887	28	10	established	establish	VERB
ejpam-5887	28	11	.	.	PUNCT
ejpam-5887	29	1	see	see	VERB
ejpam-5887	29	2	[	[	X
ejpam-5887	29	3	2	2	NUM
ejpam-5887	29	4	,	,	PUNCT
ejpam-5887	29	5	5	5	NUM
ejpam-5887	29	6	,	,	PUNCT
ejpam-5887	29	7	9	9	NUM
ejpam-5887	29	8	,	,	PUNCT
ejpam-5887	29	9	12	12	NUM
ejpam-5887	29	10	-	-	SYM
ejpam-5887	29	11	19	19	NUM
ejpam-5887	29	12	]	]	PUNCT
ejpam-5887	29	13	.	.	PUNCT
ejpam-5887	30	1	researchers	researcher	NOUN
ejpam-5887	30	2	have	have	AUX
ejpam-5887	30	3	also	also	ADV
ejpam-5887	30	4	explored	explore	VERB
ejpam-5887	30	5	the	the	DET
ejpam-5887	30	6	applications	application	NOUN
ejpam-5887	30	7	of	of	ADP
ejpam-5887	30	8	specific	specific	ADJ
ejpam-5887	30	9	graph	graph	NOUN
ejpam-5887	30	10	labeling	labeling	NOUN
ejpam-5887	30	11	techniques	technique	NOUN
ejpam-5887	30	12	,	,	PUNCT
ejpam-5887	30	13	for	for	ADP
ejpam-5887	30	14	instance	instance	NOUN
ejpam-5887	30	15	,	,	PUNCT
ejpam-5887	30	16	use	use	NOUN
ejpam-5887	30	17	of	of	ADP
ejpam-5887	30	18	‘	'	PUNCT
ejpam-5887	30	19	mean	mean	ADJ
ejpam-5887	30	20	cordial	cordial	ADJ
ejpam-5887	30	21	labeling	labeling	NOUN
ejpam-5887	30	22	’	'	PUNCT
ejpam-5887	30	23	in	in	ADP
ejpam-5887	30	24	digraph	digraph	ADJ
ejpam-5887	30	25	representations	representation	NOUN
ejpam-5887	30	26	of	of	ADP
ejpam-5887	30	27	blood	blood	NOUN
ejpam-5887	30	28	circulation	circulation	NOUN
ejpam-5887	30	29	in	in	ADP
ejpam-5887	30	30	the	the	DET
ejpam-5887	30	31	human	human	ADJ
ejpam-5887	30	32	body	body	NOUN
ejpam-5887	30	33	[	[	X
ejpam-5887	30	34	1	1	X
ejpam-5887	30	35	]	]	PUNCT
ejpam-5887	30	36	and	and	CCONJ
ejpam-5887	30	37	the	the	DET
ejpam-5887	30	38	3	3	NUM
ejpam-5887	30	39	-	-	PUNCT
ejpam-5887	30	40	total	total	ADJ
ejpam-5887	30	41	edge	edge	NOUN
ejpam-5887	30	42	product	product	NOUN
ejpam-5887	30	43	cordial	cordial	ADJ
ejpam-5887	30	44	labeling	labeling	NOUN
ejpam-5887	30	45	(	(	PUNCT
ejpam-5887	30	46	another	another	DET
ejpam-5887	30	47	variant	variant	NOUN
ejpam-5887	30	48	of	of	ADP
ejpam-5887	30	49	cordial	cordial	ADJ
ejpam-5887	30	50	labeling	labeling	NOUN
ejpam-5887	30	51	)	)	PUNCT
ejpam-5887	30	52	in	in	ADP
ejpam-5887	30	53	carbon	carbon	NOUN
ejpam-5887	30	54	nanotube	nanotube	NOUN
ejpam-5887	30	55	network	network	NOUN
ejpam-5887	30	56	[	[	X
ejpam-5887	30	57	7	7	NUM
ejpam-5887	30	58	]	]	PUNCT
ejpam-5887	30	59	.	.	PUNCT
ejpam-5887	31	1	motivated	motivate	VERB
ejpam-5887	31	2	by	by	ADP
ejpam-5887	31	3	the	the	DET
ejpam-5887	31	4	concept	concept	NOUN
ejpam-5887	31	5	of	of	ADP
ejpam-5887	31	6	‘	'	PUNCT
ejpam-5887	31	7	edge	edge	NOUN
ejpam-5887	31	8	product	product	NOUN
ejpam-5887	31	9	cordial	cordial	ADJ
ejpam-5887	31	10	labeling	labeling	NOUN
ejpam-5887	31	11	’	'	PUNCT
ejpam-5887	31	12	,	,	PUNCT
ejpam-5887	31	13	and	and	CCONJ
ejpam-5887	31	14	the	the	DET
ejpam-5887	31	15	several	several	ADJ
ejpam-5887	31	16	results	result	NOUN
ejpam-5887	31	17	established	establish	VERB
ejpam-5887	31	18	on	on	ADP
ejpam-5887	31	19	this	this	DET
ejpam-5887	31	20	concept	concept	NOUN
ejpam-5887	31	21	,	,	PUNCT
ejpam-5887	31	22	we	we	PRON
ejpam-5887	31	23	take	take	VERB
ejpam-5887	31	24	a	a	DET
ejpam-5887	31	25	step	step	NOUN
ejpam-5887	31	26	further	far	ADV
ejpam-5887	31	27	and	and	CCONJ
ejpam-5887	31	28	introduce	introduce	VERB
ejpam-5887	31	29	a	a	DET
ejpam-5887	31	30	new	new	ADJ
ejpam-5887	31	31	labeling	labeling	NOUN
ejpam-5887	31	32	namely	namely	ADV
ejpam-5887	31	33	‘	'	PUNCT
ejpam-5887	31	34	edge	edge	ADJ
ejpam-5887	31	35	k	k	ADJ
ejpam-5887	31	36	-	-	PUNCT
ejpam-5887	31	37	product	product	NOUN
ejpam-5887	31	38	cordial	cordial	ADJ
ejpam-5887	31	39	labeling	labeling	NOUN
ejpam-5887	31	40	’	'	PUNCT
ejpam-5887	31	41	as	as	SCONJ
ejpam-5887	31	42	follows	follow	VERB
ejpam-5887	31	43	:	:	PUNCT
ejpam-5887	31	44	for	for	ADP
ejpam-5887	31	45	a	a	DET
ejpam-5887	31	46	graph	graph	NOUN
ejpam-5887	31	47	g	g	NOUN
ejpam-5887	31	48	=	=	PUNCT
ejpam-5887	31	49	(	(	PUNCT
ejpam-5887	31	50	v	v	NOUN
ejpam-5887	31	51	(	(	PUNCT
ejpam-5887	31	52	g	g	NOUN
ejpam-5887	31	53	)	)	PUNCT
ejpam-5887	31	54	,	,	PUNCT
ejpam-5887	31	55	e(g	e(g	PROPN
ejpam-5887	31	56	)	)	PUNCT
ejpam-5887	31	57	)	)	PUNCT
ejpam-5887	32	1	having	have	VERB
ejpam-5887	32	2	no	no	DET
ejpam-5887	32	3	isolated	isolated	ADJ
ejpam-5887	32	4	vertex	vertex	NOUN
ejpam-5887	32	5	,	,	PUNCT
ejpam-5887	32	6	an	an	DET
ejpam-5887	32	7	edge	edge	NOUN
ejpam-5887	32	8	labeling	labeling	NOUN
ejpam-5887	33	1	f	f	NOUN
ejpam-5887	33	2	:	:	PUNCT
ejpam-5887	33	3	e(g	e(g	PROPN
ejpam-5887	33	4	)	)	PUNCT
ejpam-5887	33	5	→	→	PUNCT
ejpam-5887	33	6	{	{	PUNCT
ejpam-5887	33	7	0	0	NUM
ejpam-5887	33	8	,	,	PUNCT
ejpam-5887	33	9	1	1	NUM
ejpam-5887	33	10	,	,	PUNCT
ejpam-5887	33	11	...	...	PUNCT
ejpam-5887	33	12	,	,	PUNCT
ejpam-5887	33	13	k	k	PROPN
ejpam-5887	34	1	−	−	PROPN
ejpam-5887	34	2	1	1	NUM
ejpam-5887	34	3	}	}	PUNCT
ejpam-5887	34	4	,	,	PUNCT
ejpam-5887	34	5	where	where	SCONJ
ejpam-5887	34	6	k	k	PROPN
ejpam-5887	34	7	>	>	X
ejpam-5887	34	8	1	1	NUM
ejpam-5887	34	9	is	be	AUX
ejpam-5887	34	10	an	an	DET
ejpam-5887	34	11	integer	integer	NOUN
ejpam-5887	34	12	,	,	PUNCT
ejpam-5887	34	13	is	be	AUX
ejpam-5887	34	14	said	say	VERB
ejpam-5887	34	15	to	to	PART
ejpam-5887	34	16	be	be	AUX
ejpam-5887	34	17	an	an	DET
ejpam-5887	34	18	edge	edge	NOUN
ejpam-5887	34	19	k	k	NOUN
ejpam-5887	34	20	-	-	PUNCT
ejpam-5887	34	21	product	product	NOUN
ejpam-5887	34	22	cordial	cordial	ADJ
ejpam-5887	34	23	labeling	labeling	NOUN
ejpam-5887	34	24	if	if	SCONJ
ejpam-5887	34	25	it	it	PRON
ejpam-5887	34	26	induces	induce	VERB
ejpam-5887	34	27	a	a	DET
ejpam-5887	34	28	vertex	vertex	NOUN
ejpam-5887	34	29	labeling	labeling	NOUN
ejpam-5887	34	30	f⋆	f⋆	NUM
ejpam-5887	34	31	:	:	PUNCT
ejpam-5887	34	32	v	v	NOUN
ejpam-5887	34	33	(	(	PUNCT
ejpam-5887	34	34	g	g	NOUN
ejpam-5887	34	35	)	)	PUNCT
ejpam-5887	34	36	→	→	SYM
ejpam-5887	34	37	{	{	PUNCT
ejpam-5887	34	38	0	0	NUM
ejpam-5887	34	39	,	,	PUNCT
ejpam-5887	34	40	1	1	NUM
ejpam-5887	34	41	,	,	PUNCT
ejpam-5887	34	42	...	...	PUNCT
ejpam-5887	34	43	,	,	PUNCT
ejpam-5887	35	1	k	k	PROPN
ejpam-5887	35	2	−	−	PROPN
ejpam-5887	35	3	1	1	NUM
ejpam-5887	35	4	}	}	PUNCT
ejpam-5887	35	5	defined	define	VERB
ejpam-5887	35	6	by	by	ADP
ejpam-5887	35	7	f⋆(v	f⋆(v	NOUN
ejpam-5887	35	8	)	)	PUNCT
ejpam-5887	35	9	=	=	SYM
ejpam-5887	35	10	∏	∏	PROPN
ejpam-5887	35	11	uv∈e(g	uv∈e(g	NOUN
ejpam-5887	35	12	)	)	PUNCT
ejpam-5887	35	13	f(uv)(mod	f(uv)(mod	ADP
ejpam-5887	35	14	k	k	X
ejpam-5887	35	15	)	)	PUNCT
ejpam-5887	35	16	satisfies	satisfie	NOUN
ejpam-5887	35	17	|ef	|ef	NUM
ejpam-5887	35	18	(	(	PUNCT
ejpam-5887	35	19	i)−	i)−	PROPN
ejpam-5887	35	20	ef	ef	PROPN
ejpam-5887	35	21	(	(	PUNCT
ejpam-5887	35	22	j)|	j)|	NOUN
ejpam-5887	35	23	≤	≤	NUM
ejpam-5887	35	24	1	1	NUM
ejpam-5887	35	25	and	and	CCONJ
ejpam-5887	35	26	|vf⋆(i)−	|vf⋆(i)−	NOUN
ejpam-5887	35	27	vf⋆(j)|	vf⋆(j)|	NOUN
ejpam-5887	35	28	≤	≤	NUM
ejpam-5887	35	29	1	1	NUM
ejpam-5887	35	30	for	for	ADP
ejpam-5887	35	31	i	i	PRON
ejpam-5887	35	32	,	,	PUNCT
ejpam-5887	35	33	j	j	PROPN
ejpam-5887	35	34	∈	∈	PROPN
ejpam-5887	35	35	{	{	PUNCT
ejpam-5887	35	36	0	0	NUM
ejpam-5887	35	37	,	,	PUNCT
ejpam-5887	35	38	1	1	NUM
ejpam-5887	35	39	,	,	PUNCT
ejpam-5887	35	40	...	...	PUNCT
ejpam-5887	35	41	,	,	PUNCT
ejpam-5887	35	42	k	k	PROPN
ejpam-5887	35	43	−	−	PROPN
ejpam-5887	35	44	1	1	NUM
ejpam-5887	35	45	}	}	PUNCT
ejpam-5887	35	46	,	,	PUNCT
ejpam-5887	35	47	where	where	SCONJ
ejpam-5887	35	48	ef	ef	PROPN
ejpam-5887	35	49	(	(	PUNCT
ejpam-5887	35	50	i	i	PROPN
ejpam-5887	35	51	)	)	PUNCT
ejpam-5887	35	52	and	and	CCONJ
ejpam-5887	35	53	vf⋆(i	vf⋆(i	PROPN
ejpam-5887	35	54	)	)	PUNCT
ejpam-5887	35	55	denote	denote	VERB
ejpam-5887	35	56	the	the	DET
ejpam-5887	35	57	number	number	NOUN
ejpam-5887	35	58	of	of	ADP
ejpam-5887	35	59	edges	edge	NOUN
ejpam-5887	35	60	and	and	CCONJ
ejpam-5887	35	61	vertices	vertex	NOUN
ejpam-5887	35	62	respectively	respectively	ADV
ejpam-5887	35	63	having	have	VERB
ejpam-5887	35	64	a	a	DET
ejpam-5887	35	65	label	label	NOUN
ejpam-5887	36	1	i	i	PRON
ejpam-5887	36	2	(	(	PUNCT
ejpam-5887	36	3	i	i	NOUN
ejpam-5887	36	4	=	=	NOUN
ejpam-5887	36	5	0	0	NUM
ejpam-5887	36	6	,	,	PUNCT
ejpam-5887	36	7	1	1	NUM
ejpam-5887	36	8	,	,	PUNCT
ejpam-5887	36	9	...	...	PUNCT
ejpam-5887	36	10	,	,	PUNCT
ejpam-5887	36	11	k−1	k−1	PROPN
ejpam-5887	36	12	)	)	PUNCT
ejpam-5887	36	13	.	.	PUNCT
ejpam-5887	37	1	a	a	DET
ejpam-5887	37	2	graph	graph	NOUN
ejpam-5887	37	3	that	that	PRON
ejpam-5887	37	4	admits	admit	VERB
ejpam-5887	37	5	an	an	DET
ejpam-5887	37	6	edge	edge	NOUN
ejpam-5887	37	7	k	k	NOUN
ejpam-5887	37	8	-	-	PUNCT
ejpam-5887	37	9	product	product	NOUN
ejpam-5887	37	10	cordial	cordial	ADJ
ejpam-5887	37	11	labeling	labeling	NOUN
ejpam-5887	37	12	is	be	AUX
ejpam-5887	37	13	called	call	VERB
ejpam-5887	37	14	edge	edge	ADJ
ejpam-5887	37	15	k	k	NOUN
ejpam-5887	37	16	-	-	PUNCT
ejpam-5887	37	17	product	product	NOUN
ejpam-5887	37	18	cordial	cordial	ADJ
ejpam-5887	37	19	graph	graph	NOUN
ejpam-5887	37	20	.	.	PUNCT
ejpam-5887	38	1	in	in	ADP
ejpam-5887	38	2	this	this	DET
ejpam-5887	38	3	study	study	NOUN
ejpam-5887	38	4	,	,	PUNCT
ejpam-5887	38	5	we	we	PRON
ejpam-5887	38	6	explore	explore	VERB
ejpam-5887	38	7	the	the	DET
ejpam-5887	38	8	edge	edge	NOUN
ejpam-5887	38	9	k	k	NOUN
ejpam-5887	38	10	-	-	PUNCT
ejpam-5887	38	11	product	product	NOUN
ejpam-5887	38	12	cordial	cordial	ADJ
ejpam-5887	38	13	behavior	behavior	NOUN
ejpam-5887	38	14	of	of	ADP
ejpam-5887	38	15	some	some	DET
ejpam-5887	38	16	standard	standard	ADJ
ejpam-5887	38	17	graphs	graph	NOUN
ejpam-5887	38	18	.	.	PUNCT
ejpam-5887	39	1	we	we	PRON
ejpam-5887	39	2	present	present	VERB
ejpam-5887	39	3	our	our	PRON
ejpam-5887	39	4	study	study	NOUN
ejpam-5887	39	5	as	as	SCONJ
ejpam-5887	39	6	follows	follow	VERB
ejpam-5887	39	7	:	:	PUNCT
ejpam-5887	39	8	followed	follow	VERB
ejpam-5887	39	9	by	by	ADP
ejpam-5887	39	10	the	the	DET
ejpam-5887	39	11	introduction	introduction	NOUN
ejpam-5887	39	12	,	,	PUNCT
ejpam-5887	39	13	the	the	DET
ejpam-5887	39	14	edge	edge	NOUN
ejpam-5887	39	15	k	k	NOUN
ejpam-5887	39	16	-	-	PUNCT
ejpam-5887	39	17	product	product	NOUN
ejpam-5887	39	18	cordial	cordial	ADJ
ejpam-5887	39	19	behavior	behavior	NOUN
ejpam-5887	39	20	of	of	ADP
ejpam-5887	39	21	star	star	PROPN
ejpam-5887	39	22	,	,	PUNCT
ejpam-5887	39	23	bistar	bistar	PROPN
ejpam-5887	39	24	,	,	PUNCT
ejpam-5887	39	25	complete	complete	ADJ
ejpam-5887	39	26	graph	graph	NOUN
ejpam-5887	39	27	and	and	CCONJ
ejpam-5887	39	28	complete	complete	ADJ
ejpam-5887	39	29	bipartite	bipartite	NOUN
ejpam-5887	39	30	graph	graph	NOUN
ejpam-5887	39	31	are	be	AUX
ejpam-5887	39	32	investigated	investigate	VERB
ejpam-5887	39	33	in	in	ADP
ejpam-5887	39	34	the	the	DET
ejpam-5887	39	35	second	second	ADJ
ejpam-5887	39	36	section	section	NOUN
ejpam-5887	39	37	.	.	PUNCT
ejpam-5887	40	1	in	in	ADP
ejpam-5887	40	2	the	the	DET
ejpam-5887	40	3	third	third	ADJ
ejpam-5887	40	4	section	section	NOUN
ejpam-5887	40	5	,	,	PUNCT
ejpam-5887	40	6	we	we	PRON
ejpam-5887	40	7	focus	focus	VERB
ejpam-5887	40	8	on	on	ADP
ejpam-5887	40	9	the	the	DET
ejpam-5887	40	10	edge	edge	NOUN
ejpam-5887	40	11	k	k	NOUN
ejpam-5887	40	12	-	-	PUNCT
ejpam-5887	40	13	product	product	NOUN
ejpam-5887	40	14	cordial	cordial	ADJ
ejpam-5887	40	15	behavior	behavior	NOUN
ejpam-5887	40	16	of	of	ADP
ejpam-5887	40	17	the	the	DET
ejpam-5887	40	18	shadow	shadow	NOUN
ejpam-5887	40	19	and	and	CCONJ
ejpam-5887	40	20	splitting	splitting	NOUN
ejpam-5887	40	21	graph	graph	NOUN
ejpam-5887	40	22	of	of	ADP
ejpam-5887	40	23	star	star	NOUN
ejpam-5887	40	24	.	.	PUNCT
ejpam-5887	41	1	in	in	ADP
ejpam-5887	41	2	the	the	DET
ejpam-5887	41	3	fourth	fourth	ADJ
ejpam-5887	41	4	section	section	NOUN
ejpam-5887	41	5	,	,	PUNCT
ejpam-5887	41	6	we	we	PRON
ejpam-5887	41	7	investigate	investigate	VERB
ejpam-5887	41	8	the	the	DET
ejpam-5887	41	9	edge	edge	NOUN
ejpam-5887	41	10	k	k	NOUN
ejpam-5887	41	11	-	-	PUNCT
ejpam-5887	41	12	product	product	NOUN
ejpam-5887	41	13	cordial	cordial	ADJ
ejpam-5887	41	14	behavior	behavior	NOUN
ejpam-5887	41	15	of	of	ADP
ejpam-5887	41	16	the	the	DET
ejpam-5887	41	17	path	path	NOUN
ejpam-5887	41	18	union	union	NOUN
ejpam-5887	41	19	of	of	ADP
ejpam-5887	41	20	graphs	graph	NOUN
ejpam-5887	41	21	.	.	PUNCT
ejpam-5887	42	1	the	the	DET
ejpam-5887	42	2	definitions	definition	NOUN
ejpam-5887	42	3	of	of	ADP
ejpam-5887	42	4	the	the	DET
ejpam-5887	42	5	following	follow	VERB
ejpam-5887	42	6	graph	graph	NOUN
ejpam-5887	42	7	structures	structure	NOUN
ejpam-5887	42	8	are	be	AUX
ejpam-5887	42	9	also	also	ADV
ejpam-5887	42	10	useful	useful	ADJ
ejpam-5887	42	11	for	for	ADP
ejpam-5887	42	12	the	the	DET
ejpam-5887	42	13	present	present	ADJ
ejpam-5887	42	14	study	study	NOUN
ejpam-5887	42	15	.	.	PUNCT
ejpam-5887	43	1	definition	definition	NOUN
ejpam-5887	43	2	1	1	NUM
ejpam-5887	43	3	[	[	X
ejpam-5887	43	4	4	4	NUM
ejpam-5887	43	5	]	]	PUNCT
ejpam-5887	43	6	.	.	PUNCT
ejpam-5887	44	1	let	let	VERB
ejpam-5887	44	2	g	g	PRON
ejpam-5887	44	3	be	be	AUX
ejpam-5887	44	4	a	a	DET
ejpam-5887	44	5	graph	graph	NOUN
ejpam-5887	44	6	and	and	CCONJ
ejpam-5887	44	7	g′	g′	NOUN
ejpam-5887	44	8	be	be	AUX
ejpam-5887	44	9	a	a	DET
ejpam-5887	44	10	copy	copy	NOUN
ejpam-5887	44	11	of	of	ADP
ejpam-5887	44	12	g.	g.	PROPN
ejpam-5887	44	13	let	let	VERB
ejpam-5887	44	14	v′	v′	NOUN
ejpam-5887	44	15	be	be	AUX
ejpam-5887	44	16	the	the	DET
ejpam-5887	44	17	vertex	vertex	NOUN
ejpam-5887	44	18	in	in	ADP
ejpam-5887	44	19	g′	g′	NOUN
ejpam-5887	44	20	corresponding	corresponding	NOUN
ejpam-5887	44	21	to	to	ADP
ejpam-5887	44	22	the	the	DET
ejpam-5887	44	23	vertex	vertex	NOUN
ejpam-5887	44	24	v	v	NOUN
ejpam-5887	44	25	of	of	ADP
ejpam-5887	44	26	g.	g.	PROPN
ejpam-5887	44	27	the	the	DET
ejpam-5887	44	28	shadow	shadow	NOUN
ejpam-5887	44	29	graph	graph	NOUN
ejpam-5887	44	30	of	of	ADP
ejpam-5887	44	31	a	a	DET
ejpam-5887	44	32	graph	graph	NOUN
ejpam-5887	44	33	g	g	NOUN
ejpam-5887	44	34	,	,	PUNCT
ejpam-5887	44	35	denoted	denote	VERB
ejpam-5887	44	36	as	as	ADP
ejpam-5887	44	37	d2(g	d2(g	NOUN
ejpam-5887	44	38	)	)	PUNCT
ejpam-5887	44	39	is	be	AUX
ejpam-5887	44	40	a	a	DET
ejpam-5887	44	41	graph	graph	NOUN
ejpam-5887	44	42	obtained	obtain	VERB
ejpam-5887	44	43	by	by	ADP
ejpam-5887	44	44	the	the	DET
ejpam-5887	44	45	following	follow	VERB
ejpam-5887	44	46	operation	operation	NOUN
ejpam-5887	44	47	:	:	PUNCT
ejpam-5887	44	48	join	join	VERB
ejpam-5887	44	49	each	each	DET
ejpam-5887	44	50	vertex	vertex	NOUN
ejpam-5887	44	51	v	v	NOUN
ejpam-5887	44	52	in	in	ADP
ejpam-5887	44	53	g	g	NOUN
ejpam-5887	44	54	to	to	ADP
ejpam-5887	44	55	the	the	DET
ejpam-5887	44	56	neighbors	neighbor	NOUN
ejpam-5887	44	57	of	of	ADP
ejpam-5887	44	58	the	the	DET
ejpam-5887	44	59	vertex	vertex	NOUN
ejpam-5887	44	60	v′	v′	NOUN
ejpam-5887	44	61	in	in	ADP
ejpam-5887	44	62	g′	g′	NOUN
ejpam-5887	44	63	which	which	PRON
ejpam-5887	44	64	corresponds	correspond	VERB
ejpam-5887	44	65	to	to	ADP
ejpam-5887	44	66	v.	v.	ADP
ejpam-5887	44	67	definition	definition	NOUN
ejpam-5887	44	68	2	2	NUM
ejpam-5887	44	69	[	[	X
ejpam-5887	44	70	4	4	NUM
ejpam-5887	44	71	]	]	PUNCT
ejpam-5887	44	72	.	.	PUNCT
ejpam-5887	45	1	the	the	DET
ejpam-5887	45	2	splitting	splitting	NOUN
ejpam-5887	45	3	graph	graph	NOUN
ejpam-5887	45	4	of	of	ADP
ejpam-5887	45	5	a	a	DET
ejpam-5887	45	6	graph	graph	NOUN
ejpam-5887	45	7	g	g	NOUN
ejpam-5887	45	8	,	,	PUNCT
ejpam-5887	45	9	denoted	denote	VERB
ejpam-5887	45	10	as	as	ADP
ejpam-5887	45	11	s′(g	s′(g	PROPN
ejpam-5887	45	12	)	)	PUNCT
ejpam-5887	45	13	is	be	AUX
ejpam-5887	45	14	the	the	DET
ejpam-5887	45	15	graph	graph	NOUN
ejpam-5887	45	16	obtained	obtain	VERB
ejpam-5887	45	17	from	from	ADP
ejpam-5887	45	18	g	g	NOUN
ejpam-5887	45	19	by	by	ADP
ejpam-5887	45	20	taking	take	VERB
ejpam-5887	45	21	a	a	DET
ejpam-5887	45	22	new	new	ADJ
ejpam-5887	45	23	vertex	vertex	NOUN
ejpam-5887	45	24	u′	u′	PROPN
ejpam-5887	45	25	for	for	ADP
ejpam-5887	45	26	each	each	DET
ejpam-5887	45	27	u	u	PROPN
ejpam-5887	45	28	∈	∈	PROPN
ejpam-5887	45	29	v	v	NOUN
ejpam-5887	45	30	(	(	PUNCT
ejpam-5887	45	31	g	g	NOUN
ejpam-5887	45	32	)	)	PUNCT
ejpam-5887	45	33	and	and	CCONJ
ejpam-5887	45	34	joining	join	VERB
ejpam-5887	45	35	u′	u′	PRON
ejpam-5887	45	36	to	to	ADP
ejpam-5887	45	37	all	all	DET
ejpam-5887	45	38	vertices	vertex	NOUN
ejpam-5887	45	39	of	of	ADP
ejpam-5887	45	40	g	g	NOUN
ejpam-5887	45	41	adjacent	adjacent	ADJ
ejpam-5887	45	42	to	to	PART
ejpam-5887	45	43	u.	u.	VERB
ejpam-5887	45	44	definition	definition	NOUN
ejpam-5887	45	45	3	3	NUM
ejpam-5887	45	46	[	[	X
ejpam-5887	45	47	8	8	NUM
ejpam-5887	45	48	]	]	PUNCT
ejpam-5887	45	49	.	.	PUNCT
ejpam-5887	46	1	let	let	VERB
ejpam-5887	46	2	g1	g1	PROPN
ejpam-5887	46	3	,	,	PUNCT
ejpam-5887	46	4	g2	g2	PROPN
ejpam-5887	46	5	,	,	PUNCT
ejpam-5887	46	6	....	....	PUNCT
ejpam-5887	46	7	,	,	PUNCT
ejpam-5887	46	8	gn	gn	PROPN
ejpam-5887	46	9	,	,	PUNCT
ejpam-5887	46	10	n	n	PRON
ejpam-5887	46	11	≥	≥	NOUN
ejpam-5887	46	12	2	2	NUM
ejpam-5887	46	13	,	,	PUNCT
ejpam-5887	46	14	be	be	AUX
ejpam-5887	46	15	n	n	ADV
ejpam-5887	46	16	copies	copy	NOUN
ejpam-5887	46	17	of	of	ADP
ejpam-5887	46	18	a	a	DET
ejpam-5887	46	19	graph	graph	NOUN
ejpam-5887	46	20	g.	g.	NOUN
ejpam-5887	46	21	let	let	VERB
ejpam-5887	47	1	vi	vi	PROPN
ejpam-5887	47	2	∈	∈	PROPN
ejpam-5887	47	3	n.	n.	NOUN
ejpam-5887	47	4	m.	m.	NOUN
ejpam-5887	47	5	noureldeen	noureldeen	NOUN
ejpam-5887	47	6	et	et	PROPN
ejpam-5887	47	7	al	al	PROPN
ejpam-5887	47	8	.	.	PUNCT
ejpam-5887	47	9	/	/	SYM
ejpam-5887	47	10	eur	eur	PROPN
ejpam-5887	47	11	.	.	PUNCT
ejpam-5887	48	1	j.	j.	PROPN
ejpam-5887	48	2	pure	pure	PROPN
ejpam-5887	48	3	appl	appl	PROPN
ejpam-5887	48	4	.	.	PROPN
ejpam-5887	48	5	math	math	PROPN
ejpam-5887	48	6	,	,	PUNCT
ejpam-5887	48	7	18	18	NUM
ejpam-5887	48	8	(	(	PUNCT
ejpam-5887	48	9	2	2	NUM
ejpam-5887	48	10	)	)	PUNCT
ejpam-5887	48	11	(	(	PUNCT
ejpam-5887	48	12	2025	2025	NUM
ejpam-5887	48	13	)	)	PUNCT
ejpam-5887	48	14	,	,	PUNCT
ejpam-5887	48	15	5887	5887	NUM
ejpam-5887	48	16	3	3	NUM
ejpam-5887	48	17	of	of	ADP
ejpam-5887	48	18	21	21	NUM
ejpam-5887	48	19	v	v	NOUN
ejpam-5887	48	20	(	(	PUNCT
ejpam-5887	48	21	gi	gi	NOUN
ejpam-5887	48	22	)	)	PUNCT
ejpam-5887	48	23	,	,	PUNCT
ejpam-5887	48	24	i	i	PRON
ejpam-5887	48	25	=	=	NOUN
ejpam-5887	48	26	1	1	NUM
ejpam-5887	48	27	,	,	PUNCT
ejpam-5887	48	28	2	2	NUM
ejpam-5887	48	29	,	,	PUNCT
ejpam-5887	48	30	...	...	PUNCT
ejpam-5887	48	31	,	,	PUNCT
ejpam-5887	48	32	n	n	CCONJ
ejpam-5887	48	33	be	be	VERB
ejpam-5887	48	34	the	the	DET
ejpam-5887	48	35	vertex	vertex	NOUN
ejpam-5887	48	36	corresponding	correspond	VERB
ejpam-5887	48	37	to	to	ADP
ejpam-5887	48	38	the	the	DET
ejpam-5887	48	39	vertex	vertex	NOUN
ejpam-5887	48	40	v	v	ADP
ejpam-5887	48	41	∈	∈	PROPN
ejpam-5887	48	42	v	v	NOUN
ejpam-5887	48	43	(	(	PUNCT
ejpam-5887	48	44	g	g	NOUN
ejpam-5887	48	45	)	)	PUNCT
ejpam-5887	48	46	in	in	ADP
ejpam-5887	48	47	the	the	DET
ejpam-5887	48	48	ith	ith	PROPN
ejpam-5887	48	49	copy	copy	NOUN
ejpam-5887	48	50	of	of	ADP
ejpam-5887	48	51	gi	gi	INTJ
ejpam-5887	48	52	.	.	PUNCT
ejpam-5887	49	1	we	we	PRON
ejpam-5887	49	2	denoted	denote	VERB
ejpam-5887	49	3	by	by	ADP
ejpam-5887	49	4	p	p	PROPN
ejpam-5887	49	5	(	(	PUNCT
ejpam-5887	49	6	n.gv	n.gv	PROPN
ejpam-5887	49	7	)	)	PUNCT
ejpam-5887	49	8	the	the	DET
ejpam-5887	49	9	graph	graph	NOUN
ejpam-5887	49	10	obtained	obtain	VERB
ejpam-5887	49	11	by	by	ADP
ejpam-5887	49	12	adding	add	VERB
ejpam-5887	49	13	the	the	DET
ejpam-5887	49	14	edge	edge	NOUN
ejpam-5887	49	15	vivi+1	vivi+1	ADJ
ejpam-5887	49	16	to	to	PART
ejpam-5887	49	17	gi	gi	VERB
ejpam-5887	49	18	and	and	CCONJ
ejpam-5887	49	19	gi+1	gi+1	VERB
ejpam-5887	49	20	,	,	PUNCT
ejpam-5887	49	21	1	1	NUM
ejpam-5887	49	22	≤	≤	NUM
ejpam-5887	49	23	i	i	PRON
ejpam-5887	49	24	≤	≤	ADJ
ejpam-5887	49	25	n−	n−	PROPN
ejpam-5887	49	26	1	1	NUM
ejpam-5887	49	27	,	,	PUNCT
ejpam-5887	49	28	and	and	CCONJ
ejpam-5887	49	29	we	we	PRON
ejpam-5887	49	30	call	call	VERB
ejpam-5887	49	31	p	p	X
ejpam-5887	49	32	(	(	PUNCT
ejpam-5887	49	33	n.gv	n.gv	PROPN
ejpam-5887	49	34	)	)	PUNCT
ejpam-5887	49	35	the	the	DET
ejpam-5887	49	36	path	path	NOUN
ejpam-5887	49	37	union	union	PROPN
ejpam-5887	49	38	of	of	ADP
ejpam-5887	49	39	n	n	PROPN
ejpam-5887	49	40	copies	copy	NOUN
ejpam-5887	49	41	of	of	ADP
ejpam-5887	49	42	the	the	DET
ejpam-5887	49	43	graph	graph	NOUN
ejpam-5887	49	44	g.	g.	NOUN
ejpam-5887	49	45	2	2	NUM
ejpam-5887	49	46	.	.	PUNCT
ejpam-5887	49	47	edge	edge	PROPN
ejpam-5887	49	48	k	k	NOUN
ejpam-5887	49	49	-	-	PUNCT
ejpam-5887	49	50	product	product	NOUN
ejpam-5887	49	51	cordial	cordial	ADJ
ejpam-5887	49	52	labeling	labeling	NOUN
ejpam-5887	49	53	of	of	ADP
ejpam-5887	49	54	star	star	NOUN
ejpam-5887	49	55	,	,	PUNCT
ejpam-5887	49	56	bistar	bistar	PROPN
ejpam-5887	49	57	,	,	PUNCT
ejpam-5887	49	58	complete	complete	ADJ
ejpam-5887	49	59	graph	graph	NOUN
ejpam-5887	49	60	and	and	CCONJ
ejpam-5887	49	61	complete	complete	ADJ
ejpam-5887	49	62	bipartite	bipartite	NOUN
ejpam-5887	49	63	graph	graph	NOUN
ejpam-5887	49	64	in	in	ADP
ejpam-5887	49	65	this	this	DET
ejpam-5887	49	66	section	section	NOUN
ejpam-5887	49	67	,	,	PUNCT
ejpam-5887	49	68	first	first	ADV
ejpam-5887	49	69	we	we	PRON
ejpam-5887	49	70	establish	establish	VERB
ejpam-5887	49	71	that	that	SCONJ
ejpam-5887	49	72	the	the	DET
ejpam-5887	49	73	star	star	NOUN
ejpam-5887	49	74	graph	graph	NOUN
ejpam-5887	49	75	k1,n	k1,n	PROPN
ejpam-5887	49	76	and	and	CCONJ
ejpam-5887	49	77	the	the	DET
ejpam-5887	49	78	bistar	bistar	PROPN
ejpam-5887	49	79	graph	graph	NOUN
ejpam-5887	49	80	bn	bn	PROPN
ejpam-5887	49	81	,	,	PUNCT
ejpam-5887	49	82	n	n	PRON
ejpam-5887	49	83	admit	admit	VERB
ejpam-5887	49	84	an	an	DET
ejpam-5887	49	85	edge	edge	NOUN
ejpam-5887	49	86	k	k	NOUN
ejpam-5887	49	87	-	-	PUNCT
ejpam-5887	49	88	product	product	NOUN
ejpam-5887	49	89	cordial	cordial	ADJ
ejpam-5887	49	90	labeling	labeling	NOUN
ejpam-5887	49	91	for	for	ADP
ejpam-5887	49	92	n	n	DET
ejpam-5887	49	93	≥	≥	NOUN
ejpam-5887	49	94	k.	k.	INTJ
ejpam-5887	50	1	in	in	ADP
ejpam-5887	50	2	the	the	DET
ejpam-5887	50	3	next	next	ADJ
ejpam-5887	50	4	two	two	NUM
ejpam-5887	50	5	theorems	theorem	NOUN
ejpam-5887	50	6	,	,	PUNCT
ejpam-5887	50	7	we	we	PRON
ejpam-5887	50	8	give	give	VERB
ejpam-5887	50	9	the	the	DET
ejpam-5887	50	10	necessary	necessary	ADJ
ejpam-5887	50	11	condition	condition	NOUN
ejpam-5887	50	12	for	for	ADP
ejpam-5887	50	13	the	the	DET
ejpam-5887	50	14	complete	complete	ADJ
ejpam-5887	50	15	graph	graph	NOUN
ejpam-5887	50	16	kn	kn	PROPN
ejpam-5887	50	17	and	and	CCONJ
ejpam-5887	50	18	the	the	DET
ejpam-5887	50	19	complete	complete	ADJ
ejpam-5887	50	20	bipartite	bipartite	PROPN
ejpam-5887	50	21	graph	graph	NOUN
ejpam-5887	50	22	km	km	PROPN
ejpam-5887	50	23	,	,	PUNCT
ejpam-5887	50	24	n	n	PRON
ejpam-5887	50	25	to	to	PART
ejpam-5887	50	26	admit	admit	VERB
ejpam-5887	50	27	an	an	DET
ejpam-5887	50	28	edge	edge	NOUN
ejpam-5887	50	29	k	k	NOUN
ejpam-5887	50	30	-	-	PUNCT
ejpam-5887	50	31	product	product	NOUN
ejpam-5887	50	32	cordial	cordial	ADJ
ejpam-5887	50	33	labeling	labeling	NOUN
ejpam-5887	50	34	.	.	PUNCT
ejpam-5887	51	1	theorem	theorem	NOUN
ejpam-5887	51	2	1	1	NUM
ejpam-5887	51	3	.	.	PUNCT
ejpam-5887	51	4	for	for	ADP
ejpam-5887	51	5	n	n	PRON
ejpam-5887	51	6	≥	≥	NOUN
ejpam-5887	51	7	k	k	NOUN
ejpam-5887	51	8	,	,	PUNCT
ejpam-5887	51	9	the	the	DET
ejpam-5887	51	10	star	star	NOUN
ejpam-5887	51	11	k1,n	k1,n	PROPN
ejpam-5887	51	12	admits	admit	VERB
ejpam-5887	51	13	an	an	DET
ejpam-5887	51	14	edge	edge	NOUN
ejpam-5887	51	15	k	k	NOUN
ejpam-5887	51	16	-	-	PUNCT
ejpam-5887	51	17	product	product	NOUN
ejpam-5887	51	18	cordial	cordial	ADJ
ejpam-5887	51	19	labeling	labeling	NOUN
ejpam-5887	51	20	.	.	PUNCT
ejpam-5887	52	1	proof	proof	NOUN
ejpam-5887	52	2	.	.	PUNCT
ejpam-5887	53	1	let	let	VERB
ejpam-5887	53	2	the	the	DET
ejpam-5887	53	3	vertex	vertex	NOUN
ejpam-5887	53	4	set	set	NOUN
ejpam-5887	53	5	and	and	CCONJ
ejpam-5887	53	6	edge	edge	NOUN
ejpam-5887	53	7	set	set	NOUN
ejpam-5887	53	8	of	of	ADP
ejpam-5887	53	9	k1,n	k1,n	PROPN
ejpam-5887	53	10	be	be	AUX
ejpam-5887	53	11	v	v	ADP
ejpam-5887	53	12	(	(	PUNCT
ejpam-5887	53	13	k1,n	k1,n	PROPN
ejpam-5887	53	14	)	)	PUNCT
ejpam-5887	53	15	=	=	PRON
ejpam-5887	53	16	{	{	PUNCT
ejpam-5887	53	17	u	u	PROPN
ejpam-5887	53	18	,	,	PUNCT
ejpam-5887	53	19	ui	ui	PROPN
ejpam-5887	53	20	;	;	PUNCT
ejpam-5887	54	1	1	1	NUM
ejpam-5887	54	2	≤	≤	NUM
ejpam-5887	54	3	i	i	PRON
ejpam-5887	54	4	≤	≤	NOUN
ejpam-5887	54	5	n	n	CCONJ
ejpam-5887	54	6	}	}	PUNCT
ejpam-5887	54	7	and	and	CCONJ
ejpam-5887	54	8	e(k1,n	e(k1,n	NOUN
ejpam-5887	54	9	)	)	PUNCT
ejpam-5887	54	10	=	=	PRON
ejpam-5887	54	11	{	{	PUNCT
ejpam-5887	54	12	uui	uui	NOUN
ejpam-5887	54	13	;	;	PUNCT
ejpam-5887	54	14	1	1	NUM
ejpam-5887	54	15	≤	≤	NUM
ejpam-5887	54	16	i	i	PRON
ejpam-5887	54	17	≤	≤	NOUN
ejpam-5887	54	18	n	n	CCONJ
ejpam-5887	54	19	}	}	PUNCT
ejpam-5887	54	20	respectively	respectively	ADV
ejpam-5887	54	21	.	.	PUNCT
ejpam-5887	55	1	let	let	VERB
ejpam-5887	55	2	n	n	PRON
ejpam-5887	55	3	≡	≡	PROPN
ejpam-5887	55	4	r	r	NOUN
ejpam-5887	55	5	(	(	PUNCT
ejpam-5887	55	6	mod	mod	PROPN
ejpam-5887	55	7	k	k	PROPN
ejpam-5887	55	8	)	)	PUNCT
ejpam-5887	55	9	;	;	PUNCT
ejpam-5887	55	10	0	0	NUM
ejpam-5887	55	11	≤	≤	NUM
ejpam-5887	55	12	r	r	NOUN
ejpam-5887	55	13	≤	≤	PUNCT
ejpam-5887	55	14	k	k	NOUN
ejpam-5887	56	1	−	−	NOUN
ejpam-5887	56	2	1	1	X
ejpam-5887	56	3	.	.	PUNCT
ejpam-5887	56	4	define	define	VERB
ejpam-5887	56	5	f	f	PROPN
ejpam-5887	56	6	:	:	PUNCT
ejpam-5887	56	7	e(k1,n	e(k1,n	X
ejpam-5887	56	8	)	)	PUNCT
ejpam-5887	56	9	→	→	SYM
ejpam-5887	56	10	{	{	PUNCT
ejpam-5887	56	11	0	0	NUM
ejpam-5887	56	12	,	,	PUNCT
ejpam-5887	56	13	1	1	NUM
ejpam-5887	56	14	,	,	PUNCT
ejpam-5887	56	15	2	2	NUM
ejpam-5887	56	16	,	,	PUNCT
ejpam-5887	56	17	...	...	PUNCT
ejpam-5887	56	18	,	,	PUNCT
ejpam-5887	56	19	k	k	PROPN
ejpam-5887	57	1	−	−	PROPN
ejpam-5887	57	2	1	1	NUM
ejpam-5887	57	3	}	}	PUNCT
ejpam-5887	57	4	for	for	ADP
ejpam-5887	57	5	n	n	PRON
ejpam-5887	57	6	≥	≥	NOUN
ejpam-5887	57	7	k	k	ADV
ejpam-5887	57	8	as	as	SCONJ
ejpam-5887	57	9	follows	follow	VERB
ejpam-5887	57	10	:	:	PUNCT
ejpam-5887	57	11	f(uui	f(uui	ADJ
ejpam-5887	57	12	)	)	PUNCT
ejpam-5887	57	13	=	=	SYM
ejpam-5887	57	14	0	0	NUM
ejpam-5887	57	15	;	;	PUNCT
ejpam-5887	57	16	1	1	NUM
ejpam-5887	57	17	≤	≤	NUM
ejpam-5887	57	18	i	i	PRON
ejpam-5887	57	19	≤	≤	ADV
ejpam-5887	57	20	⌊nk	⌊nk	NOUN
ejpam-5887	57	21	⌋	⌋	NOUN
ejpam-5887	57	22	,	,	PUNCT
ejpam-5887	57	23	f(uu⌊n	f(uu⌊n	PROPN
ejpam-5887	57	24	k	k	X
ejpam-5887	57	25	⌋+i	⌋+i	PROPN
ejpam-5887	57	26	)	)	PUNCT
ejpam-5887	57	27	=	=	PRON
ejpam-5887	58	1	{	{	PUNCT
ejpam-5887	58	2	q	q	NOUN
ejpam-5887	58	3	;	;	PUNCT
ejpam-5887	58	4	i	i	PRON
ejpam-5887	58	5	≡	≡	PROPN
ejpam-5887	58	6	q	q	X
ejpam-5887	58	7	(	(	PUNCT
ejpam-5887	58	8	mod	mod	X
ejpam-5887	58	9	(	(	PUNCT
ejpam-5887	58	10	k	k	NOUN
ejpam-5887	58	11	−	−	PROPN
ejpam-5887	58	12	1	1	NUM
ejpam-5887	58	13	)	)	PUNCT
ejpam-5887	58	14	)	)	PUNCT
ejpam-5887	58	15	,	,	PUNCT
ejpam-5887	58	16	1	1	NUM
ejpam-5887	58	17	≤	≤	NUM
ejpam-5887	58	18	q	q	PROPN
ejpam-5887	58	19	≤	≤	NUM
ejpam-5887	58	20	k	k	NOUN
ejpam-5887	59	1	−	−	PROPN
ejpam-5887	60	1	2	2	NUM
ejpam-5887	61	1	k	k	NOUN
ejpam-5887	61	2	−	−	PROPN
ejpam-5887	61	3	1	1	NUM
ejpam-5887	61	4	;	;	PUNCT
ejpam-5887	61	5	i	i	PRON
ejpam-5887	61	6	≡	≡	PROPN
ejpam-5887	61	7	0	0	PUNCT
ejpam-5887	62	1	(	(	PUNCT
ejpam-5887	62	2	mod	mod	PROPN
ejpam-5887	62	3	(	(	PUNCT
ejpam-5887	62	4	k	k	NOUN
ejpam-5887	62	5	−	−	PROPN
ejpam-5887	62	6	1	1	NUM
ejpam-5887	62	7	)	)	PUNCT
ejpam-5887	62	8	)	)	PUNCT
ejpam-5887	62	9	;	;	PUNCT
ejpam-5887	62	10	1	1	NUM
ejpam-5887	62	11	≤	≤	NUM
ejpam-5887	62	12	i	i	PRON
ejpam-5887	62	13	≤	≤	NUM
ejpam-5887	62	14	n−	n−	NOUN
ejpam-5887	62	15	⌊nk	⌊nk	VERB
ejpam-5887	62	16	⌋.	⌋.	ADV
ejpam-5887	62	17	from	from	ADP
ejpam-5887	62	18	this	this	DET
ejpam-5887	62	19	labeling	labeling	NOUN
ejpam-5887	62	20	we	we	PRON
ejpam-5887	62	21	get	get	VERB
ejpam-5887	62	22	,	,	PUNCT
ejpam-5887	62	23	ef	ef	PROPN
ejpam-5887	62	24	(	(	PUNCT
ejpam-5887	62	25	i	i	NOUN
ejpam-5887	62	26	)	)	PUNCT
ejpam-5887	62	27	=	=	PRON
ejpam-5887	62	28	{	{	PUNCT
ejpam-5887	62	29	⌊nk	⌊nk	NOUN
ejpam-5887	62	30	⌋	⌋	NOUN
ejpam-5887	62	31	;	;	PUNCT
ejpam-5887	62	32	i	i	NOUN
ejpam-5887	62	33	=	=	NOUN
ejpam-5887	62	34	0	0	NUM
ejpam-5887	62	35	;	;	PUNCT
ejpam-5887	62	36	r	r	NOUN
ejpam-5887	62	37	<	<	X
ejpam-5887	62	38	i	i	NOUN
ejpam-5887	62	39	≤	≤	PUNCT
ejpam-5887	63	1	k	k	NOUN
ejpam-5887	63	2	−	−	PROPN
ejpam-5887	63	3	1	1	NUM
ejpam-5887	63	4	⌊nk	⌊nk	NOUN
ejpam-5887	63	5	⌋+	⌋+	X
ejpam-5887	63	6	1	1	NUM
ejpam-5887	63	7	;	;	PUNCT
ejpam-5887	63	8	1	1	NUM
ejpam-5887	63	9	≤	≤	NUM
ejpam-5887	63	10	i	i	X
ejpam-5887	63	11	≤	≤	ADJ
ejpam-5887	63	12	r	r	NOUN
ejpam-5887	63	13	,	,	PUNCT
ejpam-5887	63	14	vf∗(i	vf∗(i	NOUN
ejpam-5887	63	15	)	)	PUNCT
ejpam-5887	63	16	=	=	SYM
ejpam-5887	63	17	{	{	PUNCT
ejpam-5887	63	18	⌊nk	⌊nk	NOUN
ejpam-5887	63	19	⌋	⌋	NOUN
ejpam-5887	63	20	;	;	PUNCT
ejpam-5887	63	21	r	r	NOUN
ejpam-5887	63	22	<	<	X
ejpam-5887	63	23	i	i	NOUN
ejpam-5887	63	24	≤	≤	PUNCT
ejpam-5887	64	1	k	k	NOUN
ejpam-5887	64	2	−	−	PROPN
ejpam-5887	64	3	1	1	NUM
ejpam-5887	64	4	⌊nk	⌊nk	NOUN
ejpam-5887	64	5	⌋+	⌋+	X
ejpam-5887	64	6	1	1	NUM
ejpam-5887	64	7	;	;	PUNCT
ejpam-5887	64	8	0	0	NUM
ejpam-5887	64	9	≤	≤	NUM
ejpam-5887	64	10	i	i	PRON
ejpam-5887	64	11	≤	≤	PROPN
ejpam-5887	64	12	r.	r.	PROPN
ejpam-5887	64	13	clearly	clearly	ADV
ejpam-5887	64	14	,	,	PUNCT
ejpam-5887	64	15	|ef	|ef	PROPN
ejpam-5887	64	16	(	(	PUNCT
ejpam-5887	64	17	i	i	NOUN
ejpam-5887	64	18	)	)	PUNCT
ejpam-5887	64	19	−	−	PROPN
ejpam-5887	64	20	ef	ef	PROPN
ejpam-5887	64	21	(	(	PUNCT
ejpam-5887	64	22	j)|	j)|	PROPN
ejpam-5887	64	23	≤	≤	NUM
ejpam-5887	64	24	1	1	NUM
ejpam-5887	64	25	and	and	CCONJ
ejpam-5887	64	26	|vf∗(i	|vf∗(i	NUM
ejpam-5887	64	27	)	)	PUNCT
ejpam-5887	65	1	−	−	NOUN
ejpam-5887	65	2	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	65	3	≤	≤	NOUN
ejpam-5887	65	4	1	1	NUM
ejpam-5887	65	5	for	for	ADP
ejpam-5887	65	6	i	i	PRON
ejpam-5887	65	7	,	,	PUNCT
ejpam-5887	65	8	j	j	PROPN
ejpam-5887	65	9	∈	∈	PROPN
ejpam-5887	65	10	{	{	PUNCT
ejpam-5887	65	11	0	0	NUM
ejpam-5887	65	12	,	,	PUNCT
ejpam-5887	65	13	1	1	NUM
ejpam-5887	65	14	,	,	PUNCT
ejpam-5887	65	15	2	2	NUM
ejpam-5887	65	16	,	,	PUNCT
ejpam-5887	65	17	...	...	PUNCT
ejpam-5887	65	18	,	,	PUNCT
ejpam-5887	65	19	k	k	PROPN
ejpam-5887	65	20	−	−	PROPN
ejpam-5887	65	21	1	1	NUM
ejpam-5887	65	22	}	}	PUNCT
ejpam-5887	65	23	.	.	PUNCT
ejpam-5887	66	1	hence	hence	ADV
ejpam-5887	66	2	,	,	PUNCT
ejpam-5887	66	3	k1,n	k1,n	PROPN
ejpam-5887	66	4	is	be	AUX
ejpam-5887	66	5	an	an	DET
ejpam-5887	66	6	edge	edge	NOUN
ejpam-5887	66	7	k	k	NOUN
ejpam-5887	66	8	-	-	PUNCT
ejpam-5887	66	9	product	product	NOUN
ejpam-5887	66	10	cordial	cordial	ADJ
ejpam-5887	66	11	graph	graph	NOUN
ejpam-5887	66	12	for	for	ADP
ejpam-5887	66	13	n	n	PRON
ejpam-5887	66	14	≥	≥	NOUN
ejpam-5887	66	15	k.	k.	PROPN
ejpam-5887	66	16	example	example	NOUN
ejpam-5887	67	1	1	1	NUM
ejpam-5887	67	2	.	.	PUNCT
ejpam-5887	67	3	an	an	DET
ejpam-5887	67	4	edge	edge	NOUN
ejpam-5887	67	5	4	4	NUM
ejpam-5887	67	6	-	-	PUNCT
ejpam-5887	67	7	product	product	NOUN
ejpam-5887	67	8	cordial	cordial	ADJ
ejpam-5887	67	9	labeling	labeling	NOUN
ejpam-5887	67	10	of	of	ADP
ejpam-5887	67	11	k1,9	k1,9	PROPN
ejpam-5887	67	12	is	be	AUX
ejpam-5887	67	13	given	give	VERB
ejpam-5887	67	14	in	in	ADP
ejpam-5887	67	15	figure	figure	NOUN
ejpam-5887	67	16	1	1	NUM
ejpam-5887	67	17	.	.	NOUN
ejpam-5887	67	18	0	0	NUM
ejpam-5887	67	19	0	0	NUM
ejpam-5887	67	20	1	1	NUM
ejpam-5887	67	21	1	1	NUM
ejpam-5887	67	22	1	1	NUM
ejpam-5887	67	23	2	2	NUM
ejpam-5887	67	24	2	2	NUM
ejpam-5887	67	25	3	3	NUM
ejpam-5887	67	26	3	3	NUM
ejpam-5887	67	27	0	0	NUM
ejpam-5887	67	28	0	0	NUM
ejpam-5887	67	29	0	0	NUM
ejpam-5887	67	30	1	1	NUM
ejpam-5887	67	31	1	1	NUM
ejpam-5887	67	32	1	1	NUM
ejpam-5887	67	33	2	2	NUM
ejpam-5887	67	34	2	2	NUM
ejpam-5887	67	35	33	33	NUM
ejpam-5887	67	36	figure	figure	NOUN
ejpam-5887	67	37	1	1	NUM
ejpam-5887	67	38	:	:	PUNCT
ejpam-5887	67	39	edge	edge	VERB
ejpam-5887	67	40	4	4	NUM
ejpam-5887	67	41	-	-	PUNCT
ejpam-5887	67	42	product	product	NOUN
ejpam-5887	67	43	cordial	cordial	ADJ
ejpam-5887	67	44	labeling	labeling	NOUN
ejpam-5887	67	45	of	of	ADP
ejpam-5887	67	46	k1,9	k1,9	PROPN
ejpam-5887	67	47	n.	n.	PROPN
ejpam-5887	67	48	m.	m.	PROPN
ejpam-5887	67	49	noureldeen	noureldeen	INTJ
ejpam-5887	67	50	et	et	PROPN
ejpam-5887	67	51	al	al	PROPN
ejpam-5887	67	52	.	.	PUNCT
ejpam-5887	67	53	/	/	SYM
ejpam-5887	67	54	eur	eur	PROPN
ejpam-5887	67	55	.	.	PUNCT
ejpam-5887	68	1	j.	j.	PROPN
ejpam-5887	68	2	pure	pure	PROPN
ejpam-5887	68	3	appl	appl	PROPN
ejpam-5887	68	4	.	.	PROPN
ejpam-5887	68	5	math	math	PROPN
ejpam-5887	68	6	,	,	PUNCT
ejpam-5887	68	7	18	18	NUM
ejpam-5887	68	8	(	(	PUNCT
ejpam-5887	68	9	2	2	NUM
ejpam-5887	68	10	)	)	PUNCT
ejpam-5887	68	11	(	(	PUNCT
ejpam-5887	68	12	2025	2025	NUM
ejpam-5887	68	13	)	)	PUNCT
ejpam-5887	68	14	,	,	PUNCT
ejpam-5887	68	15	5887	5887	NUM
ejpam-5887	68	16	4	4	NUM
ejpam-5887	68	17	of	of	ADP
ejpam-5887	68	18	21	21	NUM
ejpam-5887	68	19	theorem	theorem	NOUN
ejpam-5887	68	20	2	2	NUM
ejpam-5887	68	21	.	.	NOUN
ejpam-5887	68	22	for	for	ADP
ejpam-5887	68	23	n	n	PRON
ejpam-5887	68	24	≥	≥	NOUN
ejpam-5887	68	25	k	k	NOUN
ejpam-5887	68	26	,	,	PUNCT
ejpam-5887	68	27	the	the	DET
ejpam-5887	68	28	bistar	bistar	PROPN
ejpam-5887	68	29	bn	bn	PROPN
ejpam-5887	68	30	,	,	PUNCT
ejpam-5887	68	31	n	n	PRON
ejpam-5887	68	32	admits	admit	VERB
ejpam-5887	68	33	an	an	DET
ejpam-5887	68	34	edge	edge	NOUN
ejpam-5887	68	35	k	k	NOUN
ejpam-5887	68	36	-	-	PUNCT
ejpam-5887	68	37	product	product	NOUN
ejpam-5887	68	38	cordial	cordial	ADJ
ejpam-5887	68	39	labeling	labeling	NOUN
ejpam-5887	68	40	.	.	PUNCT
ejpam-5887	69	1	proof	proof	NOUN
ejpam-5887	69	2	.	.	PUNCT
ejpam-5887	70	1	let	let	VERB
ejpam-5887	70	2	the	the	DET
ejpam-5887	70	3	vertex	vertex	NOUN
ejpam-5887	70	4	set	set	NOUN
ejpam-5887	70	5	and	and	CCONJ
ejpam-5887	70	6	edge	edge	NOUN
ejpam-5887	70	7	set	set	NOUN
ejpam-5887	70	8	of	of	ADP
ejpam-5887	70	9	bn	bn	NOUN
ejpam-5887	70	10	,	,	PUNCT
ejpam-5887	70	11	n	n	X
ejpam-5887	70	12	be	be	VERB
ejpam-5887	70	13	v	v	ADP
ejpam-5887	70	14	(	(	PUNCT
ejpam-5887	70	15	bn	bn	NOUN
ejpam-5887	70	16	,	,	PUNCT
ejpam-5887	70	17	n	n	CCONJ
ejpam-5887	70	18	)	)	PUNCT
ejpam-5887	70	19	=	=	PRON
ejpam-5887	70	20	{	{	PUNCT
ejpam-5887	70	21	u	u	NOUN
ejpam-5887	70	22	,	,	PUNCT
ejpam-5887	70	23	v	v	NOUN
ejpam-5887	70	24	,	,	PUNCT
ejpam-5887	70	25	ui	ui	NOUN
ejpam-5887	70	26	,	,	PUNCT
ejpam-5887	70	27	vi	vi	PROPN
ejpam-5887	70	28	;	;	PUNCT
ejpam-5887	70	29	1	1	NUM
ejpam-5887	70	30	≤	≤	NUM
ejpam-5887	70	31	i	i	PRON
ejpam-5887	70	32	≤	≤	NOUN
ejpam-5887	70	33	n	n	CCONJ
ejpam-5887	70	34	}	}	PUNCT
ejpam-5887	70	35	and	and	CCONJ
ejpam-5887	70	36	e(bn	e(bn	PROPN
ejpam-5887	70	37	,	,	PUNCT
ejpam-5887	70	38	n	n	CCONJ
ejpam-5887	70	39	)	)	PUNCT
ejpam-5887	71	1	=	=	PRON
ejpam-5887	71	2	{	{	PUNCT
ejpam-5887	71	3	uv	uv	INTJ
ejpam-5887	71	4	,	,	PUNCT
ejpam-5887	71	5	uui	uui	PROPN
ejpam-5887	71	6	,	,	PUNCT
ejpam-5887	71	7	vvi	vvi	NOUN
ejpam-5887	71	8	;	;	PUNCT
ejpam-5887	71	9	1	1	NUM
ejpam-5887	71	10	≤	≤	NUM
ejpam-5887	71	11	i	i	PRON
ejpam-5887	71	12	≤	≤	NOUN
ejpam-5887	71	13	n	n	CCONJ
ejpam-5887	71	14	}	}	PUNCT
ejpam-5887	71	15	respectively	respectively	ADV
ejpam-5887	71	16	.	.	PUNCT
ejpam-5887	72	1	let	let	VERB
ejpam-5887	72	2	n	n	PRON
ejpam-5887	72	3	≡	≡	PROPN
ejpam-5887	72	4	r	r	NOUN
ejpam-5887	72	5	(	(	PUNCT
ejpam-5887	72	6	mod	mod	PROPN
ejpam-5887	72	7	k	k	PROPN
ejpam-5887	72	8	)	)	PUNCT
ejpam-5887	72	9	;	;	PUNCT
ejpam-5887	72	10	0	0	NUM
ejpam-5887	72	11	≤	≤	NUM
ejpam-5887	72	12	r	r	NOUN
ejpam-5887	72	13	≤	≤	PUNCT
ejpam-5887	72	14	k	k	NOUN
ejpam-5887	73	1	−	−	NOUN
ejpam-5887	73	2	1	1	X
ejpam-5887	73	3	.	.	PUNCT
ejpam-5887	73	4	define	define	VERB
ejpam-5887	73	5	f	f	PROPN
ejpam-5887	73	6	:	:	PUNCT
ejpam-5887	73	7	e(bn	e(bn	PROPN
ejpam-5887	73	8	,	,	PUNCT
ejpam-5887	73	9	n	n	CCONJ
ejpam-5887	73	10	)	)	PUNCT
ejpam-5887	73	11	→	→	SYM
ejpam-5887	73	12	{	{	PUNCT
ejpam-5887	73	13	0	0	NUM
ejpam-5887	73	14	,	,	PUNCT
ejpam-5887	73	15	1	1	NUM
ejpam-5887	73	16	,	,	PUNCT
ejpam-5887	73	17	2	2	NUM
ejpam-5887	73	18	,	,	PUNCT
ejpam-5887	73	19	....	....	PUNCT
ejpam-5887	73	20	,	,	PUNCT
ejpam-5887	73	21	k	k	PROPN
ejpam-5887	74	1	−	−	PROPN
ejpam-5887	74	2	1	1	NUM
ejpam-5887	74	3	}	}	PUNCT
ejpam-5887	74	4	for	for	ADP
ejpam-5887	74	5	n	n	PRON
ejpam-5887	74	6	≥	≥	NOUN
ejpam-5887	74	7	k	k	ADV
ejpam-5887	74	8	as	as	SCONJ
ejpam-5887	74	9	follows	follow	VERB
ejpam-5887	74	10	:	:	PUNCT
ejpam-5887	74	11	f(uv	f(uv	NOUN
ejpam-5887	74	12	)	)	PUNCT
ejpam-5887	74	13	=	=	SYM
ejpam-5887	74	14	0	0	NUM
ejpam-5887	74	15	,	,	PUNCT
ejpam-5887	74	16	f(uui	f(uui	ADJ
ejpam-5887	74	17	)	)	PUNCT
ejpam-5887	74	18	=	=	SYM
ejpam-5887	74	19	0	0	NUM
ejpam-5887	74	20	;	;	PUNCT
ejpam-5887	74	21	1	1	NUM
ejpam-5887	74	22	≤	≤	NUM
ejpam-5887	74	23	i	i	PRON
ejpam-5887	74	24	≤	≤	ADV
ejpam-5887	74	25	⌊nk	⌊nk	NOUN
ejpam-5887	74	26	⌋	⌋	NOUN
ejpam-5887	74	27	,	,	PUNCT
ejpam-5887	74	28	f(uu⌊n	f(uu⌊n	PROPN
ejpam-5887	74	29	k	k	X
ejpam-5887	74	30	⌋+i	⌋+i	PROPN
ejpam-5887	74	31	)	)	PUNCT
ejpam-5887	75	1	=	=	PRON
ejpam-5887	75	2	{	{	PUNCT
ejpam-5887	75	3	q	q	NOUN
ejpam-5887	75	4	;	;	PUNCT
ejpam-5887	75	5	i	i	PRON
ejpam-5887	75	6	≡	≡	PROPN
ejpam-5887	75	7	q	q	X
ejpam-5887	76	1	(	(	PUNCT
ejpam-5887	76	2	mod	mod	X
ejpam-5887	76	3	(	(	PUNCT
ejpam-5887	76	4	k	k	NOUN
ejpam-5887	76	5	−	−	PROPN
ejpam-5887	76	6	1	1	NUM
ejpam-5887	76	7	)	)	PUNCT
ejpam-5887	76	8	)	)	PUNCT
ejpam-5887	76	9	,	,	PUNCT
ejpam-5887	76	10	1	1	NUM
ejpam-5887	76	11	≤	≤	NUM
ejpam-5887	76	12	q	q	PROPN
ejpam-5887	76	13	≤	≤	NUM
ejpam-5887	76	14	k	k	NOUN
ejpam-5887	77	1	−	−	PROPN
ejpam-5887	77	2	2	2	NUM
ejpam-5887	77	3	k	k	NOUN
ejpam-5887	77	4	−	−	PROPN
ejpam-5887	77	5	1	1	NUM
ejpam-5887	77	6	;	;	PUNCT
ejpam-5887	77	7	i	i	PRON
ejpam-5887	77	8	≡	≡	PROPN
ejpam-5887	77	9	0	0	PUNCT
ejpam-5887	78	1	(	(	PUNCT
ejpam-5887	78	2	mod	mod	PROPN
ejpam-5887	78	3	(	(	PUNCT
ejpam-5887	78	4	k	k	NOUN
ejpam-5887	78	5	−	−	PROPN
ejpam-5887	78	6	1	1	NUM
ejpam-5887	78	7	)	)	PUNCT
ejpam-5887	78	8	)	)	PUNCT
ejpam-5887	78	9	;	;	PUNCT
ejpam-5887	78	10	1	1	NUM
ejpam-5887	78	11	≤	≤	NUM
ejpam-5887	78	12	i	i	PRON
ejpam-5887	78	13	≤	≤	NUM
ejpam-5887	78	14	n−	n−	VERB
ejpam-5887	78	15	⌊nk	⌊nk	ADJ
ejpam-5887	78	16	⌋	⌋	NOUN
ejpam-5887	78	17	,	,	PUNCT
ejpam-5887	78	18	f(vvi	f(vvi	PROPN
ejpam-5887	78	19	)	)	PUNCT
ejpam-5887	78	20	=	=	SYM
ejpam-5887	78	21	0	0	NUM
ejpam-5887	78	22	;	;	PUNCT
ejpam-5887	78	23	1	1	NUM
ejpam-5887	78	24	≤	≤	NUM
ejpam-5887	78	25	i	i	PRON
ejpam-5887	78	26	≤	≤	ADV
ejpam-5887	78	27	⌊nk	⌊nk	NOUN
ejpam-5887	78	28	⌋	⌋	NOUN
ejpam-5887	78	29	−	−	PROPN
ejpam-5887	78	30	1	1	NUM
ejpam-5887	78	31	,	,	PUNCT
ejpam-5887	78	32	f(vv⌊n	f(vv⌊n	PROPN
ejpam-5887	78	33	k	k	X
ejpam-5887	78	34	⌋	⌋	PROPN
ejpam-5887	78	35	)	)	PUNCT
ejpam-5887	78	36	=	=	PUNCT
ejpam-5887	79	1			X
ejpam-5887	79	2	0	0	NUM
ejpam-5887	79	3	;	;	PUNCT
ejpam-5887	79	4	n	n	NUM
ejpam-5887	79	5	≡	≡	PROPN
ejpam-5887	79	6	1	1	NUM
ejpam-5887	79	7	,	,	PUNCT
ejpam-5887	79	8	2	2	NUM
ejpam-5887	79	9	(	(	PUNCT
ejpam-5887	79	10	mod	mod	NOUN
ejpam-5887	79	11	3	3	NUM
ejpam-5887	79	12	)	)	PUNCT
ejpam-5887	79	13	,	,	PUNCT
ejpam-5887	79	14	k	k	X
ejpam-5887	79	15	=	=	SYM
ejpam-5887	79	16	3	3	NUM
ejpam-5887	79	17	;	;	PUNCT
ejpam-5887	79	18	n	n	X
ejpam-5887	79	19	̸≡	̸≡	NOUN
ejpam-5887	79	20	0	0	NUM
ejpam-5887	79	21	,	,	PUNCT
ejpam-5887	79	22	1	1	NUM
ejpam-5887	79	23	(	(	PUNCT
ejpam-5887	79	24	mod	mod	NOUN
ejpam-5887	79	25	k	k	PROPN
ejpam-5887	79	26	)	)	PUNCT
ejpam-5887	79	27	,	,	PUNCT
ejpam-5887	79	28	k	k	X
ejpam-5887	79	29	>	>	X
ejpam-5887	79	30	3	3	NUM
ejpam-5887	79	31	1	1	NUM
ejpam-5887	79	32	;	;	PUNCT
ejpam-5887	79	33	n	n	NUM
ejpam-5887	79	34	≡	≡	PROPN
ejpam-5887	79	35	0	0	PUNCT
ejpam-5887	79	36	(	(	PUNCT
ejpam-5887	79	37	mod	mod	PROPN
ejpam-5887	79	38	k	k	PROPN
ejpam-5887	79	39	)	)	PUNCT
ejpam-5887	79	40	,	,	PUNCT
ejpam-5887	79	41	k	k	PROPN
ejpam-5887	79	42	≥	≥	NUM
ejpam-5887	79	43	3	3	NUM
ejpam-5887	79	44	k	k	NOUN
ejpam-5887	79	45	−	−	PROPN
ejpam-5887	79	46	2	2	NUM
ejpam-5887	79	47	;	;	PUNCT
ejpam-5887	79	48	k	k	PROPN
ejpam-5887	79	49	=	=	SYM
ejpam-5887	79	50	2	2	NUM
ejpam-5887	79	51	;	;	PUNCT
ejpam-5887	79	52	n	n	NUM
ejpam-5887	79	53	≡	≡	PROPN
ejpam-5887	79	54	1	1	NUM
ejpam-5887	79	55	(	(	PUNCT
ejpam-5887	79	56	mod	mod	NOUN
ejpam-5887	79	57	k	k	PROPN
ejpam-5887	79	58	)	)	PUNCT
ejpam-5887	79	59	,	,	PUNCT
ejpam-5887	79	60	k	k	X
ejpam-5887	79	61	>	>	X
ejpam-5887	79	62	3	3	NUM
ejpam-5887	79	63	,	,	PUNCT
ejpam-5887	79	64	f(vv⌊n	f(vv⌊n	X
ejpam-5887	79	65	k	k	X
ejpam-5887	79	66	⌋+i	⌋+i	PROPN
ejpam-5887	79	67	)	)	PUNCT
ejpam-5887	79	68	=	=	PRON
ejpam-5887	80	1	{	{	PUNCT
ejpam-5887	80	2	k	k	NOUN
ejpam-5887	80	3	−	−	PROPN
ejpam-5887	81	1	q	q	NOUN
ejpam-5887	81	2	;	;	PUNCT
ejpam-5887	82	1	i	i	PRON
ejpam-5887	82	2	≡	≡	PROPN
ejpam-5887	82	3	q	q	X
ejpam-5887	83	1	(	(	PUNCT
ejpam-5887	83	2	mod	mod	X
ejpam-5887	83	3	(	(	PUNCT
ejpam-5887	83	4	k	k	NOUN
ejpam-5887	83	5	−	−	PROPN
ejpam-5887	83	6	1	1	NUM
ejpam-5887	83	7	)	)	PUNCT
ejpam-5887	83	8	)	)	PUNCT
ejpam-5887	83	9	,	,	PUNCT
ejpam-5887	83	10	1	1	NUM
ejpam-5887	83	11	≤	≤	NUM
ejpam-5887	83	12	q	q	PROPN
ejpam-5887	83	13	≤	≤	NUM
ejpam-5887	83	14	k	k	NOUN
ejpam-5887	84	1	−	−	NUM
ejpam-5887	84	2	2	2	NUM
ejpam-5887	84	3	1	1	NUM
ejpam-5887	85	1	;	;	PUNCT
ejpam-5887	85	2	i	i	PRON
ejpam-5887	85	3	≡	≡	PROPN
ejpam-5887	85	4	0	0	PUNCT
ejpam-5887	86	1	(	(	PUNCT
ejpam-5887	86	2	mod	mod	PROPN
ejpam-5887	86	3	(	(	PUNCT
ejpam-5887	86	4	k	k	NOUN
ejpam-5887	86	5	−	−	PROPN
ejpam-5887	86	6	1	1	NUM
ejpam-5887	86	7	)	)	PUNCT
ejpam-5887	86	8	)	)	PUNCT
ejpam-5887	86	9	;	;	PUNCT
ejpam-5887	86	10	1	1	NUM
ejpam-5887	86	11	≤	≤	NUM
ejpam-5887	86	12	i	i	PRON
ejpam-5887	86	13	≤	≤	NUM
ejpam-5887	86	14	n−	n−	NOUN
ejpam-5887	86	15	⌊nk	⌊nk	VERB
ejpam-5887	86	16	⌋.	⌋.	ADV
ejpam-5887	86	17	from	from	ADP
ejpam-5887	86	18	this	this	DET
ejpam-5887	86	19	labeling	labeling	NOUN
ejpam-5887	86	20	we	we	PRON
ejpam-5887	86	21	obtain	obtain	VERB
ejpam-5887	86	22	,	,	PUNCT
ejpam-5887	86	23	ef	ef	PROPN
ejpam-5887	86	24	(	(	PUNCT
ejpam-5887	86	25	i	i	NOUN
ejpam-5887	86	26	)	)	PUNCT
ejpam-5887	86	27	=	=	PRON
ejpam-5887	86	28	{	{	PUNCT
ejpam-5887	86	29	2n	2n	NUM
ejpam-5887	86	30	k	k	NOUN
ejpam-5887	86	31	;	;	PUNCT
ejpam-5887	87	1	i	i	PRON
ejpam-5887	87	2	̸=	̸=	PROPN
ejpam-5887	87	3	1	1	NUM
ejpam-5887	87	4	2n	2n	NUM
ejpam-5887	87	5	k	k	NOUN
ejpam-5887	87	6	+	+	CCONJ
ejpam-5887	87	7	1	1	NUM
ejpam-5887	87	8	;	;	PUNCT
ejpam-5887	87	9	i	i	PRON
ejpam-5887	87	10	=	=	NOUN
ejpam-5887	87	11	1	1	NUM
ejpam-5887	87	12	;	;	PUNCT
ejpam-5887	87	13	n	n	NUM
ejpam-5887	87	14	≡	≡	PROPN
ejpam-5887	87	15	0	0	PUNCT
ejpam-5887	87	16	(	(	PUNCT
ejpam-5887	87	17	mod	mod	PROPN
ejpam-5887	87	18	k	k	PROPN
ejpam-5887	87	19	)	)	PUNCT
ejpam-5887	87	20	,	,	PUNCT
ejpam-5887	87	21	ef	ef	PROPN
ejpam-5887	87	22	(	(	PUNCT
ejpam-5887	87	23	i	i	NOUN
ejpam-5887	87	24	)	)	PUNCT
ejpam-5887	87	25	=	=	PRON
ejpam-5887	87	26	{	{	PUNCT
ejpam-5887	87	27	2⌊nk	2⌊nk	NUM
ejpam-5887	87	28	⌋	⌋	NOUN
ejpam-5887	87	29	;	;	PUNCT
ejpam-5887	87	30	i	i	PROPN
ejpam-5887	87	31	=	=	NOUN
ejpam-5887	87	32	0	0	NUM
ejpam-5887	87	33	,	,	PUNCT
ejpam-5887	87	34	k	k	X
ejpam-5887	87	35	>	>	X
ejpam-5887	87	36	3	3	NUM
ejpam-5887	87	37	;	;	PUNCT
ejpam-5887	87	38	2	2	NUM
ejpam-5887	87	39	≤	≤	NUM
ejpam-5887	87	40	i	i	X
ejpam-5887	87	41	≤	≤	NOUN
ejpam-5887	88	1	k	k	PRON
ejpam-5887	89	1	−	−	NUM
ejpam-5887	89	2	3	3	NUM
ejpam-5887	89	3	2⌊nk	2⌊nk	NUM
ejpam-5887	89	4	⌋+	⌋+	NUM
ejpam-5887	89	5	1	1	NUM
ejpam-5887	89	6	;	;	PUNCT
ejpam-5887	89	7	i	i	PRON
ejpam-5887	89	8	=	=	NOUN
ejpam-5887	89	9	1	1	NUM
ejpam-5887	89	10	,	,	PUNCT
ejpam-5887	89	11	k	k	PROPN
ejpam-5887	90	1	−	−	PROPN
ejpam-5887	90	2	1	1	NUM
ejpam-5887	90	3	,	,	PUNCT
ejpam-5887	90	4	k	k	PROPN
ejpam-5887	90	5	−	−	PROPN
ejpam-5887	90	6	2	2	NUM
ejpam-5887	90	7	;	;	PUNCT
ejpam-5887	90	8	i	i	PROPN
ejpam-5887	90	9	=	=	NOUN
ejpam-5887	90	10	0	0	NUM
ejpam-5887	90	11	,	,	PUNCT
ejpam-5887	90	12	k	k	PROPN
ejpam-5887	90	13	=	=	SYM
ejpam-5887	90	14	3	3	NUM
ejpam-5887	90	15	;	;	PUNCT
ejpam-5887	90	16	n	n	NUM
ejpam-5887	90	17	≡	≡	PROPN
ejpam-5887	90	18	1	1	NUM
ejpam-5887	90	19	(	(	PUNCT
ejpam-5887	90	20	mod	mod	NOUN
ejpam-5887	90	21	k	k	PROPN
ejpam-5887	90	22	)	)	PUNCT
ejpam-5887	90	23	,	,	PUNCT
ejpam-5887	90	24	k	k	PROPN
ejpam-5887	90	25	≥	≥	NUM
ejpam-5887	90	26	3	3	NUM
ejpam-5887	90	27	,	,	PUNCT
ejpam-5887	90	28	ef	ef	PROPN
ejpam-5887	90	29	(	(	PUNCT
ejpam-5887	90	30	i	i	NOUN
ejpam-5887	90	31	)	)	PUNCT
ejpam-5887	90	32	=	=	PRON
ejpam-5887	90	33	{	{	PUNCT
ejpam-5887	90	34	2⌊nk	2⌊nk	NUM
ejpam-5887	90	35	⌋+	⌋+	NUM
ejpam-5887	90	36	1	1	NUM
ejpam-5887	90	37	;	;	PUNCT
ejpam-5887	91	1	i	i	PRON
ejpam-5887	91	2	=	=	NOUN
ejpam-5887	91	3	0	0	NUM
ejpam-5887	91	4	2⌊nk	2⌊nk	NUM
ejpam-5887	91	5	⌋+	⌋+	ADJ
ejpam-5887	91	6	2	2	NUM
ejpam-5887	91	7	;	;	PUNCT
ejpam-5887	91	8	1	1	NUM
ejpam-5887	91	9	≤	≤	NUM
ejpam-5887	91	10	i	i	X
ejpam-5887	91	11	≤	≤	NOUN
ejpam-5887	92	1	k	k	PRON
ejpam-5887	93	1	−	−	PROPN
ejpam-5887	93	2	1	1	NUM
ejpam-5887	93	3	;	;	PUNCT
ejpam-5887	93	4	n	n	NUM
ejpam-5887	93	5	≡	≡	PROPN
ejpam-5887	93	6	k	k	PROPN
ejpam-5887	94	1	−	−	PROPN
ejpam-5887	94	2	1	1	NUM
ejpam-5887	94	3	(	(	PUNCT
ejpam-5887	94	4	mod	mod	PROPN
ejpam-5887	94	5	k	k	PROPN
ejpam-5887	94	6	)	)	PUNCT
ejpam-5887	94	7	,	,	PUNCT
ejpam-5887	94	8	ef	ef	PROPN
ejpam-5887	94	9	(	(	PUNCT
ejpam-5887	94	10	i	i	NOUN
ejpam-5887	94	11	)	)	PUNCT
ejpam-5887	94	12	=	=	PUNCT
ejpam-5887	95	1			PROPN
ejpam-5887	95	2	2⌊nk	2⌊nk	NUM
ejpam-5887	95	3	⌋+	⌋+	ADJ
ejpam-5887	95	4	1	1	NUM
ejpam-5887	95	5	;	;	PUNCT
ejpam-5887	95	6	i	i	PRON
ejpam-5887	95	7	=	=	NOUN
ejpam-5887	95	8	0	0	NUM
ejpam-5887	95	9	,	,	PUNCT
ejpam-5887	95	10	1	1	NUM
ejpam-5887	95	11	;	;	PUNCT
ejpam-5887	95	12	2	2	NUM
ejpam-5887	95	13	≤	≤	NUM
ejpam-5887	95	14	i	i	PRON
ejpam-5887	96	1	≤	≤	NOUN
ejpam-5887	96	2	r	r	NOUN
ejpam-5887	96	3	<	<	X
ejpam-5887	96	4	k	k	X
ejpam-5887	97	1	−	−	X
ejpam-5887	97	2	i	i	PRON
ejpam-5887	97	3	2⌊nk	2⌊nk	NUM
ejpam-5887	97	4	⌋+	⌋+	ADJ
ejpam-5887	97	5	2	2	NUM
ejpam-5887	97	6	;	;	PUNCT
ejpam-5887	97	7	i	i	PRON
ejpam-5887	97	8	≤	≤	NOUN
ejpam-5887	97	9	r	r	NOUN
ejpam-5887	97	10	,	,	PUNCT
ejpam-5887	97	11	k	k	PROPN
ejpam-5887	97	12	−	−	PROPN
ejpam-5887	98	1	i	i	PRON
ejpam-5887	98	2	≤	≤	ADJ
ejpam-5887	98	3	r	r	NOUN
ejpam-5887	98	4	;	;	PUNCT
ejpam-5887	98	5	n	n	X
ejpam-5887	98	6	̸≡	̸≡	NOUN
ejpam-5887	98	7	0	0	NUM
ejpam-5887	98	8	,	,	PUNCT
ejpam-5887	98	9	1	1	NUM
ejpam-5887	98	10	,	,	PUNCT
ejpam-5887	98	11	k	k	PROPN
ejpam-5887	98	12	−	−	PROPN
ejpam-5887	98	13	1	1	NUM
ejpam-5887	98	14	(	(	PUNCT
ejpam-5887	98	15	mod	mod	PROPN
ejpam-5887	98	16	k	k	PROPN
ejpam-5887	98	17	)	)	PUNCT
ejpam-5887	98	18	,	,	PUNCT
ejpam-5887	98	19	k	k	PROPN
ejpam-5887	98	20	≥	≥	NUM
ejpam-5887	98	21	3	3	NUM
ejpam-5887	98	22	,	,	PUNCT
ejpam-5887	98	23	;	;	PUNCT
ejpam-5887	98	24	k	k	PROPN
ejpam-5887	98	25	−	−	PROPN
ejpam-5887	99	1	i	i	PRON
ejpam-5887	99	2	≤	≤	NOUN
ejpam-5887	100	1	r	r	NOUN
ejpam-5887	100	2	<	<	X
ejpam-5887	100	3	i	i	PRON
ejpam-5887	100	4	vf∗(i	vf∗(i	ADJ
ejpam-5887	100	5	)	)	PUNCT
ejpam-5887	100	6	=	=	PRON
ejpam-5887	100	7	{	{	PUNCT
ejpam-5887	100	8	ef	ef	X
ejpam-5887	100	9	(	(	PUNCT
ejpam-5887	100	10	i	i	NOUN
ejpam-5887	100	11	)	)	PUNCT
ejpam-5887	101	1	+	+	CCONJ
ejpam-5887	101	2	1	1	NUM
ejpam-5887	101	3	;	;	PUNCT
ejpam-5887	101	4	i	i	PRON
ejpam-5887	101	5	=	=	SYM
ejpam-5887	101	6	0	0	NUM
ejpam-5887	101	7	ef	ef	PROPN
ejpam-5887	101	8	(	(	PUNCT
ejpam-5887	101	9	i	i	PROPN
ejpam-5887	101	10	)	)	PUNCT
ejpam-5887	101	11	;	;	PUNCT
ejpam-5887	101	12	1	1	NUM
ejpam-5887	101	13	≤	≤	NUM
ejpam-5887	101	14	i	i	X
ejpam-5887	101	15	≤	≤	NOUN
ejpam-5887	102	1	k	k	PRON
ejpam-5887	103	1	−	−	NOUN
ejpam-5887	103	2	1	1	X
ejpam-5887	103	3	.	.	PUNCT
ejpam-5887	104	1	clearly	clearly	ADV
ejpam-5887	104	2	,	,	PUNCT
ejpam-5887	104	3	|ef	|ef	X
ejpam-5887	104	4	(	(	PUNCT
ejpam-5887	104	5	i)−	i)−	PROPN
ejpam-5887	104	6	ef	ef	PROPN
ejpam-5887	104	7	(	(	PUNCT
ejpam-5887	104	8	j)|	j)|	NOUN
ejpam-5887	104	9	≤	≤	NUM
ejpam-5887	104	10	1	1	NUM
ejpam-5887	104	11	and	and	CCONJ
ejpam-5887	104	12	|vf⋆(i)−	|vf⋆(i)−	NOUN
ejpam-5887	104	13	vf⋆(j)|	vf⋆(j)|	NOUN
ejpam-5887	104	14	≤	≤	NUM
ejpam-5887	104	15	1	1	NUM
ejpam-5887	104	16	for	for	ADP
ejpam-5887	104	17	i	i	PRON
ejpam-5887	104	18	,	,	PUNCT
ejpam-5887	104	19	j	j	PROPN
ejpam-5887	104	20	∈	∈	PROPN
ejpam-5887	104	21	{	{	PUNCT
ejpam-5887	104	22	0	0	NUM
ejpam-5887	104	23	,	,	PUNCT
ejpam-5887	104	24	1	1	NUM
ejpam-5887	104	25	,	,	PUNCT
ejpam-5887	104	26	...	...	PUNCT
ejpam-5887	104	27	,	,	PUNCT
ejpam-5887	104	28	k	k	PROPN
ejpam-5887	105	1	−	−	PROPN
ejpam-5887	105	2	1	1	NUM
ejpam-5887	105	3	}	}	PUNCT
ejpam-5887	105	4	.	.	PUNCT
ejpam-5887	106	1	therefore	therefore	ADV
ejpam-5887	106	2	,	,	PUNCT
ejpam-5887	106	3	bn	bn	ADV
ejpam-5887	106	4	,	,	PUNCT
ejpam-5887	106	5	n	n	PRON
ejpam-5887	106	6	is	be	AUX
ejpam-5887	106	7	an	an	DET
ejpam-5887	106	8	edge	edge	NOUN
ejpam-5887	106	9	k	k	NOUN
ejpam-5887	106	10	-	-	PUNCT
ejpam-5887	106	11	product	product	NOUN
ejpam-5887	106	12	cordial	cordial	ADJ
ejpam-5887	106	13	graph	graph	NOUN
ejpam-5887	106	14	if	if	SCONJ
ejpam-5887	106	15	n	n	PRON
ejpam-5887	106	16	≥	≥	NOUN
ejpam-5887	106	17	k.	k.	PROPN
ejpam-5887	106	18	example	example	NOUN
ejpam-5887	107	1	2	2	NUM
ejpam-5887	107	2	.	.	PUNCT
ejpam-5887	107	3	an	an	DET
ejpam-5887	107	4	edge	edge	NOUN
ejpam-5887	107	5	4	4	NUM
ejpam-5887	107	6	-	-	PUNCT
ejpam-5887	107	7	product	product	NOUN
ejpam-5887	107	8	cordial	cordial	ADJ
ejpam-5887	107	9	labeling	labeling	NOUN
ejpam-5887	107	10	of	of	ADP
ejpam-5887	107	11	b9,9	b9,9	PROPN
ejpam-5887	107	12	is	be	AUX
ejpam-5887	107	13	given	give	VERB
ejpam-5887	107	14	in	in	ADP
ejpam-5887	107	15	figure	figure	NOUN
ejpam-5887	107	16	2	2	NUM
ejpam-5887	107	17	.	.	PUNCT
ejpam-5887	107	18	n.	n.	PROPN
ejpam-5887	107	19	m.	m.	PROPN
ejpam-5887	108	1	noureldeen	noureldeen	INTJ
ejpam-5887	108	2	et	et	PROPN
ejpam-5887	108	3	al	al	PROPN
ejpam-5887	108	4	.	.	PUNCT
ejpam-5887	108	5	/	/	SYM
ejpam-5887	108	6	eur	eur	PROPN
ejpam-5887	108	7	.	.	PUNCT
ejpam-5887	109	1	j.	j.	PROPN
ejpam-5887	109	2	pure	pure	PROPN
ejpam-5887	109	3	appl	appl	PROPN
ejpam-5887	109	4	.	.	PROPN
ejpam-5887	109	5	math	math	PROPN
ejpam-5887	109	6	,	,	PUNCT
ejpam-5887	109	7	18	18	NUM
ejpam-5887	109	8	(	(	PUNCT
ejpam-5887	109	9	2	2	NUM
ejpam-5887	109	10	)	)	PUNCT
ejpam-5887	109	11	(	(	PUNCT
ejpam-5887	109	12	2025	2025	NUM
ejpam-5887	109	13	)	)	PUNCT
ejpam-5887	109	14	,	,	PUNCT
ejpam-5887	109	15	5887	5887	NUM
ejpam-5887	109	16	5	5	NUM
ejpam-5887	109	17	of	of	ADP
ejpam-5887	109	18	21	21	NUM
ejpam-5887	109	19	0	0	NUM
ejpam-5887	109	20	00	00	NUM
ejpam-5887	109	21	0	0	NUM
ejpam-5887	109	22	0	0	NUM
ejpam-5887	109	23	0	0	NUM
ejpam-5887	109	24	0	0	NUM
ejpam-5887	109	25	0	0	NUM
ejpam-5887	109	26	0	0	NUM
ejpam-5887	109	27	1	1	NUM
ejpam-5887	109	28	1	1	NUM
ejpam-5887	109	29	1	1	NUM
ejpam-5887	109	30	1	1	NUM
ejpam-5887	109	31	1	1	NUM
ejpam-5887	109	32	1	1	NUM
ejpam-5887	109	33	1	1	NUM
ejpam-5887	109	34	1	1	NUM
ejpam-5887	109	35	1	1	NUM
ejpam-5887	109	36	1	1	NUM
ejpam-5887	109	37	2	2	NUM
ejpam-5887	109	38	2	2	NUM
ejpam-5887	109	39	2	2	NUM
ejpam-5887	109	40	2	2	NUM
ejpam-5887	109	41	2	2	NUM
ejpam-5887	109	42	2	2	NUM
ejpam-5887	109	43	2	2	NUM
ejpam-5887	109	44	2	2	NUM
ejpam-5887	109	45	2	2	NUM
ejpam-5887	109	46	2	2	NUM
ejpam-5887	109	47	33	33	NUM
ejpam-5887	109	48	3	3	NUM
ejpam-5887	109	49	3	3	NUM
ejpam-5887	109	50	3	3	NUM
ejpam-5887	109	51	3	3	NUM
ejpam-5887	109	52	3	3	NUM
ejpam-5887	109	53	3	3	NUM
ejpam-5887	109	54	3	3	NUM
ejpam-5887	109	55	3	3	NUM
ejpam-5887	109	56	figure	figure	NOUN
ejpam-5887	109	57	2	2	NUM
ejpam-5887	109	58	:	:	PUNCT
ejpam-5887	109	59	edge	edge	VERB
ejpam-5887	109	60	4	4	NUM
ejpam-5887	109	61	-	-	PUNCT
ejpam-5887	109	62	product	product	NOUN
ejpam-5887	109	63	cordial	cordial	ADJ
ejpam-5887	109	64	labeling	labeling	NOUN
ejpam-5887	109	65	of	of	ADP
ejpam-5887	109	66	b9,9	b9,9	PROPN
ejpam-5887	109	67	theorem	theorem	NOUN
ejpam-5887	109	68	3	3	NUM
ejpam-5887	109	69	.	.	PUNCT
ejpam-5887	110	1	a	a	DET
ejpam-5887	110	2	complete	complete	ADJ
ejpam-5887	110	3	graph	graph	NOUN
ejpam-5887	110	4	kn	kn	PROPN
ejpam-5887	110	5	does	do	AUX
ejpam-5887	110	6	not	not	PART
ejpam-5887	110	7	admit	admit	VERB
ejpam-5887	110	8	edge	edge	NOUN
ejpam-5887	110	9	k	k	NOUN
ejpam-5887	110	10	-	-	PUNCT
ejpam-5887	110	11	product	product	NOUN
ejpam-5887	110	12	cordial	cordial	ADJ
ejpam-5887	110	13	labeling	labeling	NOUN
ejpam-5887	110	14	if	if	SCONJ
ejpam-5887	110	15	3	3	NUM
ejpam-5887	110	16	≤	≤	NUM
ejpam-5887	110	17	k	k	X
ejpam-5887	110	18	≤	≤	NOUN
ejpam-5887	110	19	n(n−1	n(n−1	NUM
ejpam-5887	110	20	)	)	PUNCT
ejpam-5887	110	21	2	2	NUM
ejpam-5887	110	22	.	.	PUNCT
ejpam-5887	111	1	proof	proof	NOUN
ejpam-5887	111	2	.	.	PUNCT
ejpam-5887	112	1	let	let	VERB
ejpam-5887	112	2	3	3	NUM
ejpam-5887	112	3	≤	≤	NOUN
ejpam-5887	112	4	k	k	X
ejpam-5887	112	5	≤	≤	NOUN
ejpam-5887	112	6	n(n−1	n(n−1	NUM
ejpam-5887	112	7	)	)	PUNCT
ejpam-5887	112	8	2	2	NUM
ejpam-5887	112	9	,	,	PUNCT
ejpam-5887	112	10	then	then	ADV
ejpam-5887	112	11	⌊n(n−1	⌊n(n−1	NUM
ejpam-5887	112	12	)	)	PUNCT
ejpam-5887	112	13	2k	2k	NOUN
ejpam-5887	112	14	⌋	⌋	NOUN
ejpam-5887	112	15	≥	≥	NUM
ejpam-5887	112	16	1	1	X
ejpam-5887	112	17	.	.	PUNCT
ejpam-5887	113	1	let	let	VERB
ejpam-5887	113	2	f	f	PRON
ejpam-5887	113	3	be	be	AUX
ejpam-5887	113	4	an	an	DET
ejpam-5887	113	5	edge	edge	NOUN
ejpam-5887	113	6	k	k	NOUN
ejpam-5887	113	7	-	-	PUNCT
ejpam-5887	113	8	product	product	NOUN
ejpam-5887	113	9	cordial	cordial	ADJ
ejpam-5887	113	10	labeling	labeling	NOUN
ejpam-5887	113	11	of	of	ADP
ejpam-5887	113	12	kn	kn	PROPN
ejpam-5887	113	13	,	,	PUNCT
ejpam-5887	113	14	then	then	ADV
ejpam-5887	113	15	ef	ef	PROPN
ejpam-5887	113	16	(	(	PUNCT
ejpam-5887	113	17	i	i	NOUN
ejpam-5887	113	18	)	)	PUNCT
ejpam-5887	114	1	=	=	SYM
ejpam-5887	114	2	⌊n(n−1	⌊n(n−1	PROPN
ejpam-5887	114	3	)	)	PUNCT
ejpam-5887	114	4	2k	2k	NOUN
ejpam-5887	114	5	⌋	⌋	NOUN
ejpam-5887	114	6	or	or	CCONJ
ejpam-5887	114	7	⌊n(n−1	⌊n(n−1	NUM
ejpam-5887	114	8	)	)	PUNCT
ejpam-5887	114	9	2k	2k	NOUN
ejpam-5887	114	10	⌋+	⌋+	PUNCT
ejpam-5887	115	1	1	1	X
ejpam-5887	115	2	.	.	X
ejpam-5887	115	3	we	we	PRON
ejpam-5887	115	4	have	have	VERB
ejpam-5887	115	5	the	the	DET
ejpam-5887	115	6	following	follow	VERB
ejpam-5887	115	7	two	two	NUM
ejpam-5887	115	8	cases	case	NOUN
ejpam-5887	115	9	.	.	PUNCT
ejpam-5887	116	1	case	case	NOUN
ejpam-5887	116	2	(	(	PUNCT
ejpam-5887	116	3	i	i	NOUN
ejpam-5887	116	4	):	):	PUNCT
ejpam-5887	116	5	for	for	ADP
ejpam-5887	116	6	n	n	PRON
ejpam-5887	116	7	≤	≤	NOUN
ejpam-5887	117	1	k	k	NOUN
ejpam-5887	117	2	,	,	PUNCT
ejpam-5887	117	3	we	we	PRON
ejpam-5887	117	4	have	have	VERB
ejpam-5887	117	5	vf∗(i	vf∗(i	ADJ
ejpam-5887	117	6	)	)	PUNCT
ejpam-5887	117	7	=	=	SYM
ejpam-5887	117	8	0	0	NUM
ejpam-5887	117	9	or	or	CCONJ
ejpam-5887	117	10	1	1	NUM
ejpam-5887	117	11	.	.	PUNCT
ejpam-5887	118	1	if	if	SCONJ
ejpam-5887	118	2	ef	ef	PROPN
ejpam-5887	118	3	(	(	PUNCT
ejpam-5887	118	4	0	0	NUM
ejpam-5887	118	5	)	)	PUNCT
ejpam-5887	118	6	=	=	SYM
ejpam-5887	118	7	⌊n(n−1	⌊n(n−1	NUM
ejpam-5887	118	8	)	)	PUNCT
ejpam-5887	118	9	2k	2k	NOUN
ejpam-5887	118	10	⌋	⌋	NOUN
ejpam-5887	118	11	,	,	PUNCT
ejpam-5887	118	12	then	then	ADV
ejpam-5887	118	13	vf∗(0	vf∗(0	VERB
ejpam-5887	118	14	)	)	PUNCT
ejpam-5887	118	15	≥	≥	NOUN
ejpam-5887	118	16	⌊n(n−1	⌊n(n−1	NUM
ejpam-5887	118	17	)	)	PUNCT
ejpam-5887	118	18	2k	2k	NOUN
ejpam-5887	118	19	⌋	⌋	NOUN
ejpam-5887	118	20	+	+	CCONJ
ejpam-5887	118	21	1	1	NUM
ejpam-5887	118	22	>	>	SYM
ejpam-5887	118	23	1	1	NUM
ejpam-5887	118	24	.	.	PUNCT
ejpam-5887	118	25	therefore	therefore	ADV
ejpam-5887	118	26	,	,	PUNCT
ejpam-5887	118	27	|vf∗(0	|vf∗(0	X
ejpam-5887	118	28	)	)	PUNCT
ejpam-5887	118	29	−	−	NOUN
ejpam-5887	118	30	vf∗(j)|	vf∗(j)|	NOUN
ejpam-5887	118	31	>	>	X
ejpam-5887	118	32	1	1	NUM
ejpam-5887	118	33	for	for	ADP
ejpam-5887	118	34	some	some	PRON
ejpam-5887	118	35	j	j	NOUN
ejpam-5887	118	36	=	=	SYM
ejpam-5887	118	37	1	1	NUM
ejpam-5887	118	38	,	,	PUNCT
ejpam-5887	118	39	2	2	NUM
ejpam-5887	118	40	,	,	PUNCT
ejpam-5887	118	41	...	...	PUNCT
ejpam-5887	118	42	,	,	PUNCT
ejpam-5887	118	43	k	k	PROPN
ejpam-5887	119	1	−	−	PROPN
ejpam-5887	119	2	1	1	NUM
ejpam-5887	119	3	,	,	PUNCT
ejpam-5887	119	4	which	which	PRON
ejpam-5887	119	5	is	be	AUX
ejpam-5887	119	6	a	a	DET
ejpam-5887	119	7	contradiction	contradiction	NOUN
ejpam-5887	119	8	.	.	PUNCT
ejpam-5887	120	1	case	case	NOUN
ejpam-5887	120	2	(	(	PUNCT
ejpam-5887	120	3	ii	ii	NUM
ejpam-5887	120	4	):	):	PUNCT
ejpam-5887	120	5	for	for	ADP
ejpam-5887	120	6	n	n	PROPN
ejpam-5887	120	7	>	>	X
ejpam-5887	120	8	k	k	X
ejpam-5887	120	9	,	,	PUNCT
ejpam-5887	120	10	we	we	PRON
ejpam-5887	120	11	have	have	VERB
ejpam-5887	120	12	vf∗(i	vf∗(i	ADJ
ejpam-5887	120	13	)	)	PUNCT
ejpam-5887	121	1	=	=	SYM
ejpam-5887	121	2	⌊nk	⌊nk	NOUN
ejpam-5887	121	3	⌋	⌋	NOUN
ejpam-5887	121	4	or	or	CCONJ
ejpam-5887	121	5	⌊nk	⌊nk	NOUN
ejpam-5887	121	6	⌋	⌋	NOUN
ejpam-5887	121	7	+	+	CCONJ
ejpam-5887	122	1	1	1	X
ejpam-5887	122	2	.	.	X
ejpam-5887	123	1	if	if	SCONJ
ejpam-5887	123	2	ef	ef	PROPN
ejpam-5887	123	3	(	(	PUNCT
ejpam-5887	123	4	0	0	NUM
ejpam-5887	123	5	)	)	PUNCT
ejpam-5887	123	6	=	=	SYM
ejpam-5887	123	7	⌊n(n−1	⌊n(n−1	NUM
ejpam-5887	123	8	)	)	PUNCT
ejpam-5887	123	9	2k	2k	NOUN
ejpam-5887	123	10	⌋	⌋	NOUN
ejpam-5887	123	11	,	,	PUNCT
ejpam-5887	123	12	then	then	ADV
ejpam-5887	123	13	vf∗(0	vf∗(0	VERB
ejpam-5887	123	14	)	)	PUNCT
ejpam-5887	123	15	≥	≥	NOUN
ejpam-5887	123	16	⌊n(n−1	⌊n(n−1	NUM
ejpam-5887	123	17	)	)	PUNCT
ejpam-5887	123	18	2k	2k	NOUN
ejpam-5887	123	19	⌋+1	⌋+1	PROPN
ejpam-5887	123	20	>	>	X
ejpam-5887	123	21	⌊nk	⌊nk	PROPN
ejpam-5887	123	22	⌋+1	⌋+1	PROPN
ejpam-5887	123	23	.	.	PUNCT
ejpam-5887	124	1	therefore	therefore	ADV
ejpam-5887	124	2	,	,	PUNCT
ejpam-5887	124	3	|vf∗(0)−vf∗(j)|	|vf∗(0)−vf∗(j)|	X
ejpam-5887	124	4	>	>	SYM
ejpam-5887	124	5	1	1	NUM
ejpam-5887	124	6	for	for	ADP
ejpam-5887	124	7	some	some	PRON
ejpam-5887	124	8	j	j	NOUN
ejpam-5887	124	9	=	=	SYM
ejpam-5887	124	10	1	1	NUM
ejpam-5887	124	11	,	,	PUNCT
ejpam-5887	124	12	2	2	NUM
ejpam-5887	124	13	,	,	PUNCT
ejpam-5887	124	14	...	...	PUNCT
ejpam-5887	124	15	,	,	PUNCT
ejpam-5887	124	16	k−1	k−1	PROPN
ejpam-5887	124	17	,	,	PUNCT
ejpam-5887	124	18	which	which	PRON
ejpam-5887	124	19	is	be	AUX
ejpam-5887	124	20	a	a	DET
ejpam-5887	124	21	contradiction	contradiction	NOUN
ejpam-5887	124	22	.	.	PUNCT
ejpam-5887	125	1	hence	hence	ADV
ejpam-5887	125	2	,	,	PUNCT
ejpam-5887	125	3	kn	kn	PROPN
ejpam-5887	125	4	is	be	AUX
ejpam-5887	125	5	not	not	PART
ejpam-5887	125	6	an	an	DET
ejpam-5887	125	7	edge	edge	NOUN
ejpam-5887	125	8	k	k	NOUN
ejpam-5887	125	9	-	-	PUNCT
ejpam-5887	125	10	product	product	NOUN
ejpam-5887	125	11	cordial	cordial	ADJ
ejpam-5887	125	12	graph	graph	NOUN
ejpam-5887	125	13	if	if	SCONJ
ejpam-5887	125	14	3	3	NUM
ejpam-5887	125	15	≤	≤	NUM
ejpam-5887	125	16	k	k	X
ejpam-5887	125	17	≤	≤	NOUN
ejpam-5887	125	18	n(n−1	n(n−1	NUM
ejpam-5887	125	19	)	)	PUNCT
ejpam-5887	125	20	2	2	NUM
ejpam-5887	125	21	.	.	PUNCT
ejpam-5887	126	1	theorem	theorem	VERB
ejpam-5887	126	2	4	4	NUM
ejpam-5887	126	3	.	.	PUNCT
ejpam-5887	127	1	a	a	DET
ejpam-5887	127	2	complete	complete	ADJ
ejpam-5887	127	3	bipartite	bipartite	NOUN
ejpam-5887	127	4	graph	graph	NOUN
ejpam-5887	127	5	km	km	PROPN
ejpam-5887	127	6	,	,	PUNCT
ejpam-5887	127	7	n	n	PROPN
ejpam-5887	127	8	with	with	ADP
ejpam-5887	127	9	m	m	PROPN
ejpam-5887	127	10	≡	≡	PROPN
ejpam-5887	127	11	r1	r1	PROPN
ejpam-5887	127	12	(	(	PUNCT
ejpam-5887	127	13	mod	mod	PROPN
ejpam-5887	127	14	k	k	PROPN
ejpam-5887	127	15	)	)	PUNCT
ejpam-5887	127	16	and	and	CCONJ
ejpam-5887	127	17	n	n	CCONJ
ejpam-5887	127	18	≡	≡	PROPN
ejpam-5887	127	19	r2	r2	PROPN
ejpam-5887	127	20	(	(	PUNCT
ejpam-5887	127	21	mod	mod	PROPN
ejpam-5887	127	22	k	k	PROPN
ejpam-5887	127	23	)	)	PUNCT
ejpam-5887	127	24	does	do	AUX
ejpam-5887	127	25	not	not	PART
ejpam-5887	127	26	admit	admit	VERB
ejpam-5887	127	27	edge	edge	NOUN
ejpam-5887	127	28	k	k	NOUN
ejpam-5887	127	29	-	-	PUNCT
ejpam-5887	127	30	product	product	NOUN
ejpam-5887	127	31	cordial	cordial	ADJ
ejpam-5887	127	32	labeling	labeling	NOUN
ejpam-5887	127	33	if	if	SCONJ
ejpam-5887	127	34	r1	r1	PROPN
ejpam-5887	127	35	+	+	CCONJ
ejpam-5887	127	36	r2	r2	PROPN
ejpam-5887	127	37	<	<	X
ejpam-5887	127	38	k	k	PROPN
ejpam-5887	127	39	≤	≤	PROPN
ejpam-5887	127	40	r1r2	r1r2	X
ejpam-5887	127	41	.	.	PUNCT
ejpam-5887	128	1	proof	proof	NOUN
ejpam-5887	128	2	.	.	PUNCT
ejpam-5887	129	1	let	let	VERB
ejpam-5887	129	2	m	m	PROPN
ejpam-5887	129	3	≡	≡	PROPN
ejpam-5887	129	4	r1	r1	PROPN
ejpam-5887	129	5	(	(	PUNCT
ejpam-5887	129	6	mod	mod	PROPN
ejpam-5887	129	7	k	k	PROPN
ejpam-5887	129	8	)	)	PUNCT
ejpam-5887	129	9	and	and	CCONJ
ejpam-5887	129	10	n	n	CCONJ
ejpam-5887	129	11	≡	≡	PROPN
ejpam-5887	129	12	r2	r2	PROPN
ejpam-5887	129	13	(	(	PUNCT
ejpam-5887	129	14	mod	mod	PROPN
ejpam-5887	129	15	k	k	PROPN
ejpam-5887	129	16	)	)	PUNCT
ejpam-5887	129	17	.	.	PUNCT
ejpam-5887	130	1	then	then	ADV
ejpam-5887	130	2	|v	|v	PROPN
ejpam-5887	130	3	(	(	PUNCT
ejpam-5887	130	4	km	km	PROPN
ejpam-5887	130	5	,	,	PUNCT
ejpam-5887	130	6	n)|	n)|	NOUN
ejpam-5887	130	7	=	=	SYM
ejpam-5887	130	8	k(⌊mk	k(⌊mk	PROPN
ejpam-5887	130	9	⌋	⌋	NOUN
ejpam-5887	130	10	+	+	CCONJ
ejpam-5887	130	11	⌊nk	⌊nk	NOUN
ejpam-5887	130	12	⌋	⌋	NOUN
ejpam-5887	130	13	)	)	PUNCT
ejpam-5887	131	1	+	+	CCONJ
ejpam-5887	131	2	r1	r1	NOUN
ejpam-5887	131	3	+	+	CCONJ
ejpam-5887	131	4	r2	r2	PROPN
ejpam-5887	131	5	and	and	CCONJ
ejpam-5887	131	6	|e(km	|e(km	PROPN
ejpam-5887	131	7	,	,	PUNCT
ejpam-5887	131	8	n)|	n)|	NOUN
ejpam-5887	131	9	=	=	SYM
ejpam-5887	131	10	k(k⌊mk	k(k⌊mk	PROPN
ejpam-5887	131	11	⌋⌊	⌋⌊	SYM
ejpam-5887	131	12	n	n	CCONJ
ejpam-5887	131	13	k	k	NOUN
ejpam-5887	131	14	⌋+	⌋+	X
ejpam-5887	132	1	r2⌊mk	r2⌊mk	NOUN
ejpam-5887	132	2	⌋+	⌋+	VERB
ejpam-5887	132	3	r1⌊nk	r1⌊nk	PROPN
ejpam-5887	132	4	⌋	⌋	NOUN
ejpam-5887	132	5	)	)	PUNCT
ejpam-5887	133	1	+	+	CCONJ
ejpam-5887	134	1	r1r2	r1r2	X
ejpam-5887	134	2	.	.	PUNCT
ejpam-5887	135	1	let	let	VERB
ejpam-5887	135	2	f	f	PRON
ejpam-5887	135	3	be	be	AUX
ejpam-5887	135	4	an	an	DET
ejpam-5887	135	5	edge	edge	NOUN
ejpam-5887	135	6	k	k	NOUN
ejpam-5887	135	7	-	-	PUNCT
ejpam-5887	135	8	product	product	NOUN
ejpam-5887	135	9	cordial	cordial	ADJ
ejpam-5887	135	10	labeling	labeling	NOUN
ejpam-5887	135	11	of	of	ADP
ejpam-5887	135	12	km	km	PROPN
ejpam-5887	135	13	,	,	PUNCT
ejpam-5887	135	14	n.	n.	PROPN
ejpam-5887	135	15	then	then	ADV
ejpam-5887	135	16	ef	ef	PROPN
ejpam-5887	135	17	(	(	PUNCT
ejpam-5887	135	18	i	i	NOUN
ejpam-5887	135	19	)	)	PUNCT
ejpam-5887	135	20	=	=	SYM
ejpam-5887	135	21	k⌊mk	k⌊mk	X
ejpam-5887	135	22	⌋⌊	⌋⌊	SYM
ejpam-5887	135	23	n	n	CCONJ
ejpam-5887	135	24	k	k	NOUN
ejpam-5887	135	25	⌋	⌋	NOUN
ejpam-5887	136	1	+	+	CCONJ
ejpam-5887	136	2	r2⌊mk	r2⌊mk	PROPN
ejpam-5887	136	3	⌋	⌋	NOUN
ejpam-5887	136	4	+	+	CCONJ
ejpam-5887	136	5	r1⌊nk	r1⌊nk	NOUN
ejpam-5887	136	6	⌋+	⌋+	X
ejpam-5887	136	7	⌊	⌊	VERB
ejpam-5887	136	8	r1r2k	r1r2k	NUM
ejpam-5887	136	9	⌋	⌋	NOUN
ejpam-5887	136	10	or	or	CCONJ
ejpam-5887	136	11	k⌊mk	k⌊mk	X
ejpam-5887	136	12	⌋⌊	⌋⌊	PUNCT
ejpam-5887	136	13	n	n	CCONJ
ejpam-5887	136	14	k	k	NOUN
ejpam-5887	136	15	⌋+	⌋+	X
ejpam-5887	137	1	r2⌊mk	r2⌊mk	NOUN
ejpam-5887	137	2	⌋+	⌋+	VERB
ejpam-5887	137	3	r1⌊nk	r1⌊nk	VERB
ejpam-5887	137	4	⌋+	⌋+	X
ejpam-5887	137	5	⌊	⌊	VERB
ejpam-5887	137	6	r1r2k	r1r2k	NUM
ejpam-5887	137	7	⌋+1	⌋+1	NUM
ejpam-5887	137	8	and	and	CCONJ
ejpam-5887	137	9	vf∗(i	vf∗(i	ADJ
ejpam-5887	137	10	)	)	PUNCT
ejpam-5887	137	11	=	=	SYM
ejpam-5887	137	12	⌊mk	⌊mk	NOUN
ejpam-5887	137	13	⌋+	⌋+	X
ejpam-5887	137	14	⌊nk	⌊nk	ADV
ejpam-5887	137	15	⌋+	⌋+	PUNCT
ejpam-5887	138	1	⌊	⌊	VERB
ejpam-5887	138	2	r1+r2	r1+r2	PROPN
ejpam-5887	138	3	k	k	PROPN
ejpam-5887	138	4	⌋	⌋	PROPN
ejpam-5887	138	5	or	or	CCONJ
ejpam-5887	138	6	⌊mk	⌊mk	X
ejpam-5887	138	7	⌋+	⌋+	VERB
ejpam-5887	138	8	⌊nk	⌊nk	ADV
ejpam-5887	138	9	⌋+	⌋+	PUNCT
ejpam-5887	139	1	⌊	⌊	VERB
ejpam-5887	139	2	r1+r2	r1+r2	PROPN
ejpam-5887	139	3	k	k	PROPN
ejpam-5887	139	4	⌋+	⌋+	PUNCT
ejpam-5887	140	1	1	1	X
ejpam-5887	140	2	.	.	PUNCT
ejpam-5887	140	3	since	since	SCONJ
ejpam-5887	140	4	r1	r1	PROPN
ejpam-5887	140	5	+	+	CCONJ
ejpam-5887	140	6	r2	r2	PROPN
ejpam-5887	140	7	<	<	X
ejpam-5887	140	8	k	k	PROPN
ejpam-5887	140	9	≤	≤	PROPN
ejpam-5887	140	10	r1r2	r1r2	VERB
ejpam-5887	140	11	,	,	PUNCT
ejpam-5887	140	12	we	we	PRON
ejpam-5887	140	13	have	have	AUX
ejpam-5887	140	14	⌊	⌊	VERB
ejpam-5887	140	15	r1+r2	r1+r2	PROPN
ejpam-5887	140	16	k	k	PROPN
ejpam-5887	140	17	⌋	⌋	NOUN
ejpam-5887	140	18	<	<	X
ejpam-5887	140	19	⌊	⌊	PROPN
ejpam-5887	140	20	r1r2k	r1r2k	NUM
ejpam-5887	140	21	⌋.	⌋.	ADV
ejpam-5887	141	1	now	now	ADV
ejpam-5887	141	2	,	,	PUNCT
ejpam-5887	141	3	ef	ef	PROPN
ejpam-5887	141	4	(	(	PUNCT
ejpam-5887	141	5	0	0	NUM
ejpam-5887	141	6	)	)	PUNCT
ejpam-5887	141	7	=	=	NOUN
ejpam-5887	141	8	k⌊mk	k⌊mk	X
ejpam-5887	141	9	⌋⌊	⌋⌊	SYM
ejpam-5887	141	10	n	n	CCONJ
ejpam-5887	141	11	k	k	NOUN
ejpam-5887	141	12	⌋	⌋	NOUN
ejpam-5887	142	1	+	+	CCONJ
ejpam-5887	142	2	r2⌊mk	r2⌊mk	PROPN
ejpam-5887	142	3	⌋	⌋	NOUN
ejpam-5887	142	4	+	+	CCONJ
ejpam-5887	142	5	r1⌊nk	r1⌊nk	PROPN
ejpam-5887	142	6	⌋	⌋	NOUN
ejpam-5887	142	7	+	+	CCONJ
ejpam-5887	142	8	⌊	⌊	X
ejpam-5887	142	9	r1r2k	r1r2k	NUM
ejpam-5887	142	10	⌋	⌋	NOUN
ejpam-5887	142	11	implies	imply	VERB
ejpam-5887	142	12	vf∗(0	vf∗(0	NOUN
ejpam-5887	142	13	)	)	PUNCT
ejpam-5887	142	14	≥	≥	NOUN
ejpam-5887	142	15	k⌊mk	k⌊mk	X
ejpam-5887	142	16	⌋⌊	⌋⌊	PUNCT
ejpam-5887	142	17	n	n	CCONJ
ejpam-5887	142	18	k	k	NOUN
ejpam-5887	142	19	⌋	⌋	NOUN
ejpam-5887	143	1	+	+	CCONJ
ejpam-5887	143	2	r2⌊mk	r2⌊mk	PROPN
ejpam-5887	143	3	⌋	⌋	NOUN
ejpam-5887	143	4	+	+	CCONJ
ejpam-5887	143	5	r1⌊nk	r1⌊nk	PROPN
ejpam-5887	143	6	⌋	⌋	NOUN
ejpam-5887	143	7	+	+	CCONJ
ejpam-5887	143	8	⌊	⌊	VERB
ejpam-5887	143	9	r1r2k	r1r2k	NUM
ejpam-5887	143	10	⌋	⌋	NOUN
ejpam-5887	143	11	+	+	CCONJ
ejpam-5887	143	12	1	1	X
ejpam-5887	143	13	>	>	PUNCT
ejpam-5887	143	14	⌊mk	⌊mk	NOUN
ejpam-5887	143	15	⌋	⌋	NOUN
ejpam-5887	143	16	+	+	CCONJ
ejpam-5887	143	17	⌊nk	⌊nk	ADJ
ejpam-5887	143	18	⌋	⌋	NOUN
ejpam-5887	143	19	+	+	CCONJ
ejpam-5887	143	20	⌊	⌊	PROPN
ejpam-5887	143	21	r1+r2	r1+r2	PROPN
ejpam-5887	143	22	k	k	NOUN
ejpam-5887	143	23	⌋+1	⌋+1	PROPN
ejpam-5887	143	24	,	,	PUNCT
ejpam-5887	143	25	which	which	PRON
ejpam-5887	143	26	is	be	AUX
ejpam-5887	143	27	a	a	DET
ejpam-5887	143	28	contradiction	contradiction	NOUN
ejpam-5887	143	29	.	.	PUNCT
ejpam-5887	144	1	hence	hence	ADV
ejpam-5887	144	2	,	,	PUNCT
ejpam-5887	144	3	km	km	PROPN
ejpam-5887	144	4	,	,	PUNCT
ejpam-5887	144	5	n	n	PRON
ejpam-5887	144	6	is	be	AUX
ejpam-5887	144	7	not	not	PART
ejpam-5887	144	8	an	an	DET
ejpam-5887	144	9	edge	edge	NOUN
ejpam-5887	144	10	k	k	NOUN
ejpam-5887	144	11	-	-	PUNCT
ejpam-5887	144	12	product	product	NOUN
ejpam-5887	144	13	cordial	cordial	ADJ
ejpam-5887	144	14	graph	graph	NOUN
ejpam-5887	144	15	if	if	SCONJ
ejpam-5887	144	16	r1	r1	PROPN
ejpam-5887	144	17	+	+	CCONJ
ejpam-5887	144	18	r2	r2	PROPN
ejpam-5887	144	19	<	<	X
ejpam-5887	144	20	k	k	PROPN
ejpam-5887	144	21	≤	≤	PROPN
ejpam-5887	144	22	r1r2	r1r2	X
ejpam-5887	144	23	.	.	PUNCT
ejpam-5887	145	1	n.	n.	PROPN
ejpam-5887	145	2	m.	m.	PROPN
ejpam-5887	145	3	noureldeen	noureldeen	INTJ
ejpam-5887	145	4	et	et	PROPN
ejpam-5887	145	5	al	al	PROPN
ejpam-5887	145	6	.	.	PUNCT
ejpam-5887	145	7	/	/	SYM
ejpam-5887	145	8	eur	eur	PROPN
ejpam-5887	145	9	.	.	PUNCT
ejpam-5887	146	1	j.	j.	PROPN
ejpam-5887	146	2	pure	pure	PROPN
ejpam-5887	146	3	appl	appl	PROPN
ejpam-5887	146	4	.	.	PROPN
ejpam-5887	146	5	math	math	PROPN
ejpam-5887	146	6	,	,	PUNCT
ejpam-5887	146	7	18	18	NUM
ejpam-5887	146	8	(	(	PUNCT
ejpam-5887	146	9	2	2	NUM
ejpam-5887	146	10	)	)	PUNCT
ejpam-5887	146	11	(	(	PUNCT
ejpam-5887	146	12	2025	2025	NUM
ejpam-5887	146	13	)	)	PUNCT
ejpam-5887	146	14	,	,	PUNCT
ejpam-5887	146	15	5887	5887	NUM
ejpam-5887	146	16	6	6	NUM
ejpam-5887	146	17	of	of	ADP
ejpam-5887	146	18	21	21	NUM
ejpam-5887	146	19	3	3	NUM
ejpam-5887	146	20	.	.	PUNCT
ejpam-5887	147	1	edge	edge	PROPN
ejpam-5887	147	2	k	k	NOUN
ejpam-5887	147	3	-	-	PUNCT
ejpam-5887	147	4	product	product	NOUN
ejpam-5887	147	5	cordial	cordial	ADJ
ejpam-5887	147	6	behavior	behavior	NOUN
ejpam-5887	147	7	of	of	ADP
ejpam-5887	147	8	shadow	shadow	NOUN
ejpam-5887	147	9	and	and	CCONJ
ejpam-5887	147	10	splitting	splitting	NOUN
ejpam-5887	147	11	graph	graph	NOUN
ejpam-5887	147	12	of	of	ADP
ejpam-5887	147	13	star	star	NOUN
ejpam-5887	147	14	in	in	ADP
ejpam-5887	147	15	order	order	NOUN
ejpam-5887	147	16	to	to	PART
ejpam-5887	147	17	prove	prove	VERB
ejpam-5887	147	18	the	the	DET
ejpam-5887	147	19	edge	edge	NOUN
ejpam-5887	147	20	k	k	NOUN
ejpam-5887	147	21	-	-	PUNCT
ejpam-5887	147	22	product	product	NOUN
ejpam-5887	147	23	cordial	cordial	ADJ
ejpam-5887	147	24	behavior	behavior	NOUN
ejpam-5887	147	25	of	of	ADP
ejpam-5887	147	26	shadow	shadow	NOUN
ejpam-5887	147	27	and	and	CCONJ
ejpam-5887	147	28	splitting	splitting	NOUN
ejpam-5887	147	29	graph	graph	NOUN
ejpam-5887	147	30	of	of	ADP
ejpam-5887	147	31	star	star	NOUN
ejpam-5887	147	32	,	,	PUNCT
ejpam-5887	147	33	we	we	PRON
ejpam-5887	147	34	prove	prove	VERB
ejpam-5887	147	35	the	the	DET
ejpam-5887	147	36	following	follow	VERB
ejpam-5887	147	37	general	general	ADJ
ejpam-5887	147	38	result	result	NOUN
ejpam-5887	147	39	.	.	PUNCT
ejpam-5887	148	1	theorem	theorem	ADJ
ejpam-5887	148	2	5	5	NUM
ejpam-5887	148	3	.	.	PUNCT
ejpam-5887	149	1	a	a	DET
ejpam-5887	149	2	graph	graph	NOUN
ejpam-5887	149	3	g	g	NOUN
ejpam-5887	149	4	with	with	ADP
ejpam-5887	149	5	k	k	PROPN
ejpam-5887	149	6	≤	≤	PROPN
ejpam-5887	149	7	|v	|v	VERB
ejpam-5887	149	8	|	|	ADV
ejpam-5887	149	9	≤	≤	NUM
ejpam-5887	149	10	|e|	|e|	PRON
ejpam-5887	149	11	does	do	AUX
ejpam-5887	149	12	not	not	PART
ejpam-5887	149	13	admit	admit	VERB
ejpam-5887	149	14	an	an	DET
ejpam-5887	149	15	edge	edge	NOUN
ejpam-5887	149	16	k	k	NOUN
ejpam-5887	149	17	-	-	PUNCT
ejpam-5887	149	18	product	product	NOUN
ejpam-5887	149	19	cordial	cordial	ADJ
ejpam-5887	149	20	labeling	labeling	NOUN
ejpam-5887	149	21	if	if	SCONJ
ejpam-5887	149	22	|v	|v	PROPN
ejpam-5887	149	23	|	|	ADV
ejpam-5887	149	24	≡	≡	PROPN
ejpam-5887	149	25	0(mod	0(mod	NOUN
ejpam-5887	149	26	k	k	NOUN
ejpam-5887	149	27	)	)	PUNCT
ejpam-5887	149	28	.	.	PUNCT
ejpam-5887	150	1	proof	proof	NOUN
ejpam-5887	150	2	.	.	PUNCT
ejpam-5887	151	1	let	let	VERB
ejpam-5887	151	2	g	g	PRON
ejpam-5887	151	3	be	be	AUX
ejpam-5887	151	4	a	a	DET
ejpam-5887	151	5	graph	graph	NOUN
ejpam-5887	151	6	with	with	ADP
ejpam-5887	151	7	k	k	PROPN
ejpam-5887	151	8	≤	≤	PROPN
ejpam-5887	151	9	|v	|v	VERB
ejpam-5887	151	10	|	|	ADV
ejpam-5887	151	11	≤	≤	NUM
ejpam-5887	151	12	|e|	|e|	PRON
ejpam-5887	151	13	and	and	CCONJ
ejpam-5887	151	14	|v	|v	ADJ
ejpam-5887	151	15	|	|	ADV
ejpam-5887	151	16	=	=	SYM
ejpam-5887	151	17	tk	tk	PROPN
ejpam-5887	151	18	(	(	PUNCT
ejpam-5887	151	19	t	t	PROPN
ejpam-5887	151	20	≥	≥	PROPN
ejpam-5887	151	21	1	1	NUM
ejpam-5887	151	22	)	)	PUNCT
ejpam-5887	151	23	.	.	PUNCT
ejpam-5887	152	1	then	then	ADV
ejpam-5887	152	2	|e|	|e|	PROPN
ejpam-5887	152	3	=	=	PUNCT
ejpam-5887	152	4	tk	tk	PROPN
ejpam-5887	152	5	+	+	PROPN
ejpam-5887	152	6	j	j	PROPN
ejpam-5887	152	7	,	,	PUNCT
ejpam-5887	152	8	where	where	SCONJ
ejpam-5887	152	9	0	0	NUM
ejpam-5887	152	10	≤	≤	NUM
ejpam-5887	152	11	j	j	PROPN
ejpam-5887	152	12	≤	≤	NUM
ejpam-5887	152	13	⌊	⌊	VERB
ejpam-5887	152	14	tk(tk−3	tk(tk−3	NUM
ejpam-5887	152	15	)	)	PUNCT
ejpam-5887	152	16	2	2	NUM
ejpam-5887	153	1	⌋.	⌋.	ADV
ejpam-5887	153	2	let	let	VERB
ejpam-5887	153	3	f	f	PRON
ejpam-5887	153	4	be	be	AUX
ejpam-5887	153	5	an	an	DET
ejpam-5887	153	6	edge	edge	NOUN
ejpam-5887	153	7	k	k	NOUN
ejpam-5887	153	8	-	-	PUNCT
ejpam-5887	153	9	product	product	NOUN
ejpam-5887	153	10	cordial	cordial	ADJ
ejpam-5887	153	11	labeling	labeling	NOUN
ejpam-5887	153	12	of	of	ADP
ejpam-5887	153	13	g.	g.	PROPN
ejpam-5887	153	14	then	then	ADV
ejpam-5887	153	15	,	,	PUNCT
ejpam-5887	153	16	ef	ef	PROPN
ejpam-5887	153	17	(	(	PUNCT
ejpam-5887	153	18	i	i	NOUN
ejpam-5887	153	19	)	)	PUNCT
ejpam-5887	153	20	is	be	AUX
ejpam-5887	153	21	either	either	PRON
ejpam-5887	153	22	t+	t+	VERB
ejpam-5887	153	23	⌊	⌊	PROPN
ejpam-5887	153	24	j	j	PROPN
ejpam-5887	153	25	k⌋	k⌋	PROPN
ejpam-5887	153	26	or	or	CCONJ
ejpam-5887	153	27	t+	t+	VERB
ejpam-5887	153	28	⌊	⌊	PROPN
ejpam-5887	153	29	j	j	PROPN
ejpam-5887	154	1	k⌋+	k⌋+	NUM
ejpam-5887	154	2	1	1	NUM
ejpam-5887	154	3	and	and	CCONJ
ejpam-5887	154	4	vf∗(i	vf∗(i	ADJ
ejpam-5887	154	5	)	)	PUNCT
ejpam-5887	154	6	=	=	SYM
ejpam-5887	154	7	t	t	PROPN
ejpam-5887	154	8	(	(	PUNCT
ejpam-5887	154	9	i	i	NOUN
ejpam-5887	154	10	=	=	NOUN
ejpam-5887	154	11	0	0	NUM
ejpam-5887	154	12	,	,	PUNCT
ejpam-5887	154	13	1	1	NUM
ejpam-5887	154	14	,	,	PUNCT
ejpam-5887	154	15	...	...	PUNCT
ejpam-5887	154	16	,	,	PUNCT
ejpam-5887	154	17	k	k	PROPN
ejpam-5887	155	1	−	−	PROPN
ejpam-5887	155	2	1	1	NUM
ejpam-5887	155	3	)	)	PUNCT
ejpam-5887	155	4	.	.	PUNCT
ejpam-5887	156	1	if	if	SCONJ
ejpam-5887	156	2	ef	ef	PROPN
ejpam-5887	156	3	(	(	PUNCT
ejpam-5887	156	4	0	0	NUM
ejpam-5887	156	5	)	)	PUNCT
ejpam-5887	156	6	=	=	VERB
ejpam-5887	156	7	t+	t+	PUNCT
ejpam-5887	156	8	⌊	⌊	PROPN
ejpam-5887	157	1	j	j	PROPN
ejpam-5887	157	2	k⌋	k⌋	PROPN
ejpam-5887	157	3	,	,	PUNCT
ejpam-5887	157	4	then	then	ADV
ejpam-5887	157	5	vf∗(0	vf∗(0	VERB
ejpam-5887	157	6	)	)	PUNCT
ejpam-5887	157	7	≥	≥	NOUN
ejpam-5887	157	8	t+	t+	VERB
ejpam-5887	157	9	⌊	⌊	PROPN
ejpam-5887	157	10	j	j	PROPN
ejpam-5887	157	11	k⌋+	k⌋+	PROPN
ejpam-5887	157	12	1	1	NUM
ejpam-5887	157	13	>	>	SYM
ejpam-5887	157	14	t	t	PROPN
ejpam-5887	157	15	,	,	PUNCT
ejpam-5887	157	16	which	which	PRON
ejpam-5887	157	17	is	be	AUX
ejpam-5887	157	18	not	not	PART
ejpam-5887	157	19	possible	possible	ADJ
ejpam-5887	157	20	.	.	PUNCT
ejpam-5887	158	1	hence	hence	ADV
ejpam-5887	158	2	,	,	PUNCT
ejpam-5887	158	3	ef	ef	PROPN
ejpam-5887	158	4	(	(	PUNCT
ejpam-5887	158	5	0	0	NUM
ejpam-5887	158	6	)	)	PUNCT
ejpam-5887	158	7	=	=	VERB
ejpam-5887	158	8	t+	t+	PUNCT
ejpam-5887	158	9	⌊	⌊	PROPN
ejpam-5887	158	10	j	j	PROPN
ejpam-5887	158	11	k⌋+	k⌋+	PROPN
ejpam-5887	158	12	1	1	NUM
ejpam-5887	158	13	,	,	PUNCT
ejpam-5887	158	14	which	which	PRON
ejpam-5887	158	15	implies	imply	VERB
ejpam-5887	158	16	vf∗(0	vf∗(0	NOUN
ejpam-5887	158	17	)	)	PUNCT
ejpam-5887	158	18	≥	≥	NOUN
ejpam-5887	158	19	t	t	NOUN
ejpam-5887	158	20	+	+	CCONJ
ejpam-5887	158	21	⌊	⌊	X
ejpam-5887	158	22	j	j	NOUN
ejpam-5887	158	23	k⌋	k⌋	NOUN
ejpam-5887	158	24	+	+	CCONJ
ejpam-5887	158	25	2	2	NUM
ejpam-5887	158	26	>	>	PUNCT
ejpam-5887	158	27	t.	t.	NOUN
ejpam-5887	158	28	then	then	ADV
ejpam-5887	158	29	we	we	PRON
ejpam-5887	158	30	get	get	VERB
ejpam-5887	158	31	,	,	PUNCT
ejpam-5887	158	32	|vf⋆(0)−	|vf⋆(0)−	NOUN
ejpam-5887	158	33	vf⋆(j)|	vf⋆(j)|	SYM
ejpam-5887	158	34	>	>	SYM
ejpam-5887	158	35	1	1	NUM
ejpam-5887	158	36	for	for	ADP
ejpam-5887	158	37	some	some	PRON
ejpam-5887	158	38	j	j	NOUN
ejpam-5887	158	39	=	=	SYM
ejpam-5887	158	40	1	1	NUM
ejpam-5887	158	41	,	,	PUNCT
ejpam-5887	158	42	2	2	NUM
ejpam-5887	158	43	,	,	PUNCT
ejpam-5887	158	44	...	...	PUNCT
ejpam-5887	158	45	,	,	PUNCT
ejpam-5887	158	46	k	k	PROPN
ejpam-5887	159	1	−	−	PROPN
ejpam-5887	159	2	1	1	NUM
ejpam-5887	159	3	,	,	PUNCT
ejpam-5887	159	4	that	that	PRON
ejpam-5887	159	5	is	be	AUX
ejpam-5887	159	6	a	a	DET
ejpam-5887	159	7	contradiction	contradiction	NOUN
ejpam-5887	159	8	.	.	PUNCT
ejpam-5887	160	1	therefore	therefore	ADV
ejpam-5887	160	2	,	,	PUNCT
ejpam-5887	160	3	g	g	PROPN
ejpam-5887	160	4	is	be	AUX
ejpam-5887	160	5	not	not	PART
ejpam-5887	160	6	an	an	DET
ejpam-5887	160	7	edge	edge	NOUN
ejpam-5887	160	8	k	k	NOUN
ejpam-5887	160	9	-	-	PUNCT
ejpam-5887	160	10	product	product	NOUN
ejpam-5887	160	11	cordial	cordial	ADJ
ejpam-5887	160	12	graph	graph	NOUN
ejpam-5887	160	13	.	.	PUNCT
ejpam-5887	161	1	3.1	3.1	NUM
ejpam-5887	161	2	.	.	PUNCT
ejpam-5887	162	1	shadow	shadow	NOUN
ejpam-5887	162	2	graph	graph	NOUN
ejpam-5887	162	3	of	of	ADP
ejpam-5887	162	4	star	star	NOUN
ejpam-5887	162	5	in	in	ADP
ejpam-5887	162	6	this	this	DET
ejpam-5887	162	7	subsection	subsection	NOUN
ejpam-5887	162	8	,	,	PUNCT
ejpam-5887	162	9	we	we	PRON
ejpam-5887	162	10	establish	establish	VERB
ejpam-5887	162	11	that	that	SCONJ
ejpam-5887	162	12	the	the	DET
ejpam-5887	162	13	shadow	shadow	NOUN
ejpam-5887	162	14	graph	graph	NOUN
ejpam-5887	162	15	of	of	ADP
ejpam-5887	162	16	a	a	DET
ejpam-5887	162	17	star	star	NOUN
ejpam-5887	162	18	graph	graph	NOUN
ejpam-5887	162	19	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	162	20	)	)	PUNCT
ejpam-5887	162	21	does	do	AUX
ejpam-5887	162	22	not	not	PART
ejpam-5887	162	23	admit	admit	VERB
ejpam-5887	162	24	the	the	DET
ejpam-5887	162	25	edge	edge	NOUN
ejpam-5887	162	26	k	k	NOUN
ejpam-5887	162	27	-	-	PUNCT
ejpam-5887	162	28	product	product	NOUN
ejpam-5887	162	29	cordial	cordial	ADJ
ejpam-5887	162	30	labeling	labeling	NOUN
ejpam-5887	162	31	for	for	ADP
ejpam-5887	162	32	k	k	PROPN
ejpam-5887	162	33	≤	≤	PROPN
ejpam-5887	162	34	n.	n.	NOUN
ejpam-5887	162	35	in	in	ADP
ejpam-5887	162	36	addition	addition	NOUN
ejpam-5887	162	37	,	,	PUNCT
ejpam-5887	162	38	we	we	PRON
ejpam-5887	162	39	investigate	investigate	VERB
ejpam-5887	162	40	the	the	DET
ejpam-5887	162	41	edge	edge	NOUN
ejpam-5887	162	42	k	k	NOUN
ejpam-5887	162	43	-	-	PUNCT
ejpam-5887	162	44	product	product	NOUN
ejpam-5887	162	45	cordial	cordial	ADJ
ejpam-5887	162	46	behavior	behavior	NOUN
ejpam-5887	162	47	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	162	48	)	)	PUNCT
ejpam-5887	162	49	for	for	ADP
ejpam-5887	162	50	k	k	PROPN
ejpam-5887	162	51	=	=	SYM
ejpam-5887	162	52	3	3	NUM
ejpam-5887	162	53	,	,	PUNCT
ejpam-5887	162	54	4	4	NUM
ejpam-5887	162	55	,	,	PUNCT
ejpam-5887	162	56	5	5	NUM
ejpam-5887	162	57	.	.	X
ejpam-5887	162	58	theorem	theorem	NOUN
ejpam-5887	162	59	6	6	NUM
ejpam-5887	162	60	.	.	PUNCT
ejpam-5887	163	1	the	the	DET
ejpam-5887	163	2	graph	graph	NOUN
ejpam-5887	163	3	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	163	4	)	)	PUNCT
ejpam-5887	163	5	does	do	AUX
ejpam-5887	163	6	not	not	PART
ejpam-5887	163	7	admit	admit	VERB
ejpam-5887	163	8	an	an	DET
ejpam-5887	163	9	edge	edge	NOUN
ejpam-5887	163	10	k	k	NOUN
ejpam-5887	163	11	-	-	PUNCT
ejpam-5887	163	12	product	product	NOUN
ejpam-5887	163	13	cordial	cordial	ADJ
ejpam-5887	163	14	labeling	labeling	NOUN
ejpam-5887	163	15	for	for	ADP
ejpam-5887	163	16	k	k	PROPN
ejpam-5887	163	17	≤	≤	PROPN
ejpam-5887	163	18	n.	n.	NOUN
ejpam-5887	163	19	proof	proof	NOUN
ejpam-5887	163	20	.	.	PUNCT
ejpam-5887	164	1	let	let	VERB
ejpam-5887	164	2	k	k	PROPN
ejpam-5887	164	3	≤	≤	PROPN
ejpam-5887	164	4	n.	n.	NOUN
ejpam-5887	164	5	we	we	PRON
ejpam-5887	164	6	consider	consider	VERB
ejpam-5887	164	7	the	the	DET
ejpam-5887	164	8	following	follow	VERB
ejpam-5887	164	9	two	two	NUM
ejpam-5887	164	10	cases	case	NOUN
ejpam-5887	164	11	.	.	PUNCT
ejpam-5887	165	1	case	case	NOUN
ejpam-5887	165	2	(	(	PUNCT
ejpam-5887	165	3	i	i	NOUN
ejpam-5887	165	4	):	):	PUNCT
ejpam-5887	165	5	for	for	ADP
ejpam-5887	165	6	n	n	NOUN
ejpam-5887	165	7	=	=	SYM
ejpam-5887	165	8	tk	tk	PROPN
ejpam-5887	166	1	+	+	CCONJ
ejpam-5887	166	2	k	k	PROPN
ejpam-5887	167	1	−	−	PROPN
ejpam-5887	167	2	1	1	NUM
ejpam-5887	167	3	,	,	PUNCT
ejpam-5887	167	4	we	we	PRON
ejpam-5887	167	5	have	have	AUX
ejpam-5887	167	6	|v	|v	VERB
ejpam-5887	167	7	|	|	ADV
ejpam-5887	168	1	=	=	SYM
ejpam-5887	168	2	2tk	2tk	PROPN
ejpam-5887	169	1	+	+	CCONJ
ejpam-5887	169	2	2k	2k	NUM
ejpam-5887	169	3	and	and	CCONJ
ejpam-5887	169	4	|e|	|e|	ADJ
ejpam-5887	169	5	=	=	X
ejpam-5887	169	6	4tk	4tk	NOUN
ejpam-5887	170	1	+	+	CCONJ
ejpam-5887	170	2	4(k	4(k	NUM
ejpam-5887	170	3	−	−	NOUN
ejpam-5887	170	4	1	1	NUM
ejpam-5887	170	5	)	)	PUNCT
ejpam-5887	170	6	.	.	PUNCT
ejpam-5887	171	1	clearly	clearly	ADV
ejpam-5887	171	2	,	,	PUNCT
ejpam-5887	171	3	|v	|v	PROPN
ejpam-5887	171	4	|	|	ADV
ejpam-5887	171	5	≡	≡	PROPN
ejpam-5887	171	6	0	0	PUNCT
ejpam-5887	172	1	(	(	PUNCT
ejpam-5887	172	2	mod	mod	PROPN
ejpam-5887	172	3	k	k	PROPN
ejpam-5887	172	4	)	)	PUNCT
ejpam-5887	173	1	and	and	CCONJ
ejpam-5887	173	2	k	k	X
ejpam-5887	173	3	<	<	X
ejpam-5887	173	4	|v	|v	PROPN
ejpam-5887	174	1	|	|	ADV
ejpam-5887	174	2	<	<	X
ejpam-5887	174	3	|e|	|e|	PROPN
ejpam-5887	174	4	.	.	PUNCT
ejpam-5887	174	5	by	by	ADP
ejpam-5887	174	6	theorem	theorem	ADJ
ejpam-5887	174	7	5	5	NUM
ejpam-5887	174	8	,	,	PUNCT
ejpam-5887	174	9	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	174	10	)	)	PUNCT
ejpam-5887	174	11	is	be	AUX
ejpam-5887	174	12	not	not	PART
ejpam-5887	174	13	an	an	DET
ejpam-5887	174	14	edge	edge	NOUN
ejpam-5887	174	15	k	k	NOUN
ejpam-5887	174	16	-	-	PUNCT
ejpam-5887	174	17	product	product	NOUN
ejpam-5887	174	18	cordial	cordial	ADJ
ejpam-5887	174	19	graph	graph	NOUN
ejpam-5887	174	20	.	.	PUNCT
ejpam-5887	175	1	case	case	NOUN
ejpam-5887	175	2	(	(	PUNCT
ejpam-5887	175	3	ii	ii	NUM
ejpam-5887	175	4	):	):	PUNCT
ejpam-5887	175	5	for	for	ADP
ejpam-5887	175	6	n	n	NOUN
ejpam-5887	175	7	=	=	SYM
ejpam-5887	175	8	tk	tk	PROPN
ejpam-5887	176	1	+	+	CCONJ
ejpam-5887	176	2	r	r	NOUN
ejpam-5887	176	3	;	;	PUNCT
ejpam-5887	176	4	t	t	PROPN
ejpam-5887	176	5	≥	≥	NUM
ejpam-5887	176	6	1	1	NUM
ejpam-5887	176	7	,	,	PUNCT
ejpam-5887	176	8	0	0	NUM
ejpam-5887	176	9	≤	≤	NUM
ejpam-5887	176	10	r	r	NOUN
ejpam-5887	176	11	≤	≤	NUM
ejpam-5887	176	12	k	k	NOUN
ejpam-5887	177	1	−	−	PROPN
ejpam-5887	177	2	2	2	NUM
ejpam-5887	177	3	,	,	PUNCT
ejpam-5887	177	4	we	we	PRON
ejpam-5887	177	5	have	have	AUX
ejpam-5887	177	6	|v	|v	VERB
ejpam-5887	177	7	|	|	ADV
ejpam-5887	177	8	=	=	SYM
ejpam-5887	177	9	2tk	2tk	PROPN
ejpam-5887	178	1	+	+	CCONJ
ejpam-5887	178	2	2(r	2(r	NUM
ejpam-5887	178	3	+	+	NUM
ejpam-5887	178	4	1	1	NUM
ejpam-5887	178	5	)	)	PUNCT
ejpam-5887	178	6	and	and	CCONJ
ejpam-5887	178	7	|e|	|e|	PRON
ejpam-5887	178	8	=	=	X
ejpam-5887	178	9	4tk	4tk	NOUN
ejpam-5887	178	10	+	+	CCONJ
ejpam-5887	178	11	4r	4r	NOUN
ejpam-5887	178	12	.	.	PUNCT
ejpam-5887	179	1	if	if	SCONJ
ejpam-5887	179	2	f	f	PROPN
ejpam-5887	179	3	is	be	AUX
ejpam-5887	179	4	an	an	DET
ejpam-5887	179	5	edge	edge	NOUN
ejpam-5887	179	6	k	k	NOUN
ejpam-5887	179	7	-	-	PUNCT
ejpam-5887	179	8	product	product	NOUN
ejpam-5887	179	9	cordial	cordial	ADJ
ejpam-5887	179	10	labeling	labeling	NOUN
ejpam-5887	179	11	of	of	ADP
ejpam-5887	179	12	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	179	13	)	)	PUNCT
ejpam-5887	179	14	,	,	PUNCT
ejpam-5887	179	15	then	then	ADV
ejpam-5887	179	16	ef	ef	PROPN
ejpam-5887	179	17	(	(	PUNCT
ejpam-5887	179	18	i	i	NOUN
ejpam-5887	179	19	)	)	PUNCT
ejpam-5887	179	20	=	=	PRON
ejpam-5887	179	21	{	{	PUNCT
ejpam-5887	179	22	4	4	NUM
ejpam-5887	179	23	t	t	NOUN
ejpam-5887	179	24	;	;	PUNCT
ejpam-5887	179	25	r	r	NOUN
ejpam-5887	179	26	=	=	SYM
ejpam-5887	179	27	0	0	NUM
ejpam-5887	179	28	4t+	4t+	NOUN
ejpam-5887	179	29	⌊4rk	⌊4rk	NOUN
ejpam-5887	179	30	⌋	⌋	NOUN
ejpam-5887	179	31	or	or	CCONJ
ejpam-5887	179	32	4t+	4t+	NUM
ejpam-5887	179	33	⌊4rk	⌊4rk	NOUN
ejpam-5887	179	34	⌋+	⌋+	X
ejpam-5887	179	35	1	1	NUM
ejpam-5887	179	36	;	;	PUNCT
ejpam-5887	179	37	1	1	NUM
ejpam-5887	179	38	≤	≤	NOUN
ejpam-5887	179	39	r	r	NOUN
ejpam-5887	179	40	≤	≤	PUNCT
ejpam-5887	179	41	k	k	NOUN
ejpam-5887	179	42	−	−	PROPN
ejpam-5887	179	43	2	2	NUM
ejpam-5887	179	44	,	,	PUNCT
ejpam-5887	179	45	vf∗(i	vf∗(i	ADJ
ejpam-5887	179	46	)	)	PUNCT
ejpam-5887	179	47	=	=	PUNCT
ejpam-5887	180	1			PUNCT
ejpam-5887	180	2	2t+	2t+	NUM
ejpam-5887	180	3	1	1	NUM
ejpam-5887	180	4	;	;	PUNCT
ejpam-5887	180	5	r	r	NOUN
ejpam-5887	180	6	=	=	SYM
ejpam-5887	180	7	0	0	NUM
ejpam-5887	180	8	,	,	PUNCT
ejpam-5887	180	9	k	k	PROPN
ejpam-5887	180	10	=	=	SYM
ejpam-5887	180	11	2	2	NUM
ejpam-5887	180	12	2	2	NUM
ejpam-5887	180	13	t	t	NOUN
ejpam-5887	180	14	or	or	CCONJ
ejpam-5887	180	15	2t+	2t+	NUM
ejpam-5887	180	16	1	1	NUM
ejpam-5887	180	17	;	;	PUNCT
ejpam-5887	180	18	r	r	NOUN
ejpam-5887	180	19	=	=	SYM
ejpam-5887	180	20	0	0	NUM
ejpam-5887	180	21	,	,	PUNCT
ejpam-5887	180	22	k	k	X
ejpam-5887	180	23	≥	≥	NUM
ejpam-5887	180	24	3	3	NUM
ejpam-5887	180	25	2t+	2t+	NUM
ejpam-5887	180	26	⌊2r+2	⌊2r+2	X
ejpam-5887	181	1	k	k	X
ejpam-5887	181	2	⌋	⌋	NOUN
ejpam-5887	181	3	or	or	CCONJ
ejpam-5887	181	4	2t+	2t+	NUM
ejpam-5887	181	5	⌊2r+2	⌊2r+2	X
ejpam-5887	182	1	k	k	X
ejpam-5887	182	2	⌋+	⌋+	ADJ
ejpam-5887	182	3	1	1	NUM
ejpam-5887	182	4	;	;	PUNCT
ejpam-5887	182	5	1	1	NUM
ejpam-5887	182	6	≤	≤	NOUN
ejpam-5887	182	7	r	r	NOUN
ejpam-5887	182	8	≤	≤	PUNCT
ejpam-5887	182	9	k	k	NOUN
ejpam-5887	182	10	−	−	PROPN
ejpam-5887	182	11	2	2	X
ejpam-5887	182	12	.	.	X
ejpam-5887	182	13	for	for	ADP
ejpam-5887	182	14	the	the	DET
ejpam-5887	182	15	case	case	NOUN
ejpam-5887	182	16	where	where	SCONJ
ejpam-5887	182	17	r	r	NOUN
ejpam-5887	182	18	=	=	SYM
ejpam-5887	182	19	0	0	NUM
ejpam-5887	182	20	,	,	PUNCT
ejpam-5887	182	21	ef	ef	X
ejpam-5887	182	22	(	(	PUNCT
ejpam-5887	182	23	0	0	NUM
ejpam-5887	182	24	)	)	PUNCT
ejpam-5887	182	25	=	=	SYM
ejpam-5887	182	26	4	4	NUM
ejpam-5887	182	27	t	t	NOUN
ejpam-5887	182	28	implies	imply	VERB
ejpam-5887	182	29	vf∗(0	vf∗(0	NOUN
ejpam-5887	182	30	)	)	PUNCT
ejpam-5887	182	31	≥	≥	NOUN
ejpam-5887	182	32	4t+1	4t+1	PROPN
ejpam-5887	182	33	>	>	X
ejpam-5887	183	1	2t+1	2t+1	PROPN
ejpam-5887	183	2	for	for	ADP
ejpam-5887	183	3	t	t	PROPN
ejpam-5887	183	4	≥	≥	NUM
ejpam-5887	183	5	1	1	NUM
ejpam-5887	183	6	.	.	PUNCT
ejpam-5887	184	1	therefore	therefore	ADV
ejpam-5887	184	2	,	,	PUNCT
ejpam-5887	184	3	|vf∗(0)−vf∗(j)|	|vf∗(0)−vf∗(j)|	X
ejpam-5887	184	4	>	>	SYM
ejpam-5887	184	5	1	1	NUM
ejpam-5887	184	6	for	for	ADP
ejpam-5887	184	7	some	some	PRON
ejpam-5887	184	8	j	j	NOUN
ejpam-5887	184	9	=	=	SYM
ejpam-5887	184	10	1	1	NUM
ejpam-5887	184	11	,	,	PUNCT
ejpam-5887	184	12	2	2	NUM
ejpam-5887	184	13	,	,	PUNCT
ejpam-5887	184	14	...	...	PUNCT
ejpam-5887	184	15	,	,	PUNCT
ejpam-5887	184	16	k−1	k−1	PROPN
ejpam-5887	184	17	,	,	PUNCT
ejpam-5887	184	18	which	which	PRON
ejpam-5887	184	19	is	be	AUX
ejpam-5887	184	20	a	a	DET
ejpam-5887	184	21	contradiction	contradiction	NOUN
ejpam-5887	184	22	.	.	PUNCT
ejpam-5887	185	1	in	in	ADP
ejpam-5887	185	2	the	the	DET
ejpam-5887	185	3	other	other	ADJ
ejpam-5887	185	4	cases	case	NOUN
ejpam-5887	185	5	,	,	PUNCT
ejpam-5887	185	6	if	if	SCONJ
ejpam-5887	185	7	ef	ef	X
ejpam-5887	185	8	(	(	PUNCT
ejpam-5887	185	9	0	0	NUM
ejpam-5887	185	10	)	)	PUNCT
ejpam-5887	185	11	=	=	SYM
ejpam-5887	185	12	4	4	NUM
ejpam-5887	185	13	t	t	NOUN
ejpam-5887	185	14	+	+	NUM
ejpam-5887	185	15	⌊4rk	⌊4rk	NOUN
ejpam-5887	185	16	⌋	⌋	NOUN
ejpam-5887	185	17	,	,	PUNCT
ejpam-5887	185	18	then	then	ADV
ejpam-5887	185	19	vf∗(0	vf∗(0	VERB
ejpam-5887	185	20	)	)	PUNCT
ejpam-5887	185	21	≥	≥	NOUN
ejpam-5887	185	22	4	4	NUM
ejpam-5887	185	23	t	t	NOUN
ejpam-5887	185	24	+	+	NUM
ejpam-5887	185	25	⌊4rk	⌊4rk	NOUN
ejpam-5887	185	26	⌋	⌋	NOUN
ejpam-5887	185	27	+	+	CCONJ
ejpam-5887	185	28	1	1	X
ejpam-5887	185	29	.	.	PUNCT
ejpam-5887	186	1	since	since	SCONJ
ejpam-5887	186	2	⌊4rk	⌊4rk	NOUN
ejpam-5887	186	3	⌋	⌋	NOUN
ejpam-5887	186	4	≥	≥	PRON
ejpam-5887	186	5	⌊2r+2	⌊2r+2	X
ejpam-5887	187	1	k	k	X
ejpam-5887	187	2	⌋	⌋	NOUN
ejpam-5887	187	3	for	for	ADP
ejpam-5887	187	4	1	1	NUM
ejpam-5887	187	5	≤	≤	NOUN
ejpam-5887	187	6	r	r	NOUN
ejpam-5887	187	7	≤	≤	NUM
ejpam-5887	187	8	k	k	NOUN
ejpam-5887	188	1	−	−	PROPN
ejpam-5887	188	2	2	2	NUM
ejpam-5887	188	3	,	,	PUNCT
ejpam-5887	188	4	we	we	PRON
ejpam-5887	188	5	get	get	VERB
ejpam-5887	188	6	vf∗(0	vf∗(0	NOUN
ejpam-5887	188	7	)	)	PUNCT
ejpam-5887	188	8	>	>	X
ejpam-5887	188	9	2	2	NUM
ejpam-5887	188	10	t	t	NOUN
ejpam-5887	188	11	+	+	CCONJ
ejpam-5887	188	12	⌊2r+2	⌊2r+2	X
ejpam-5887	189	1	k	k	X
ejpam-5887	189	2	⌋	⌋	NOUN
ejpam-5887	190	1	+	+	CCONJ
ejpam-5887	190	2	1	1	NUM
ejpam-5887	190	3	for	for	ADP
ejpam-5887	190	4	t	t	PROPN
ejpam-5887	190	5	≥	≥	NUM
ejpam-5887	190	6	1	1	NUM
ejpam-5887	190	7	.	.	PUNCT
ejpam-5887	191	1	therefore	therefore	ADV
ejpam-5887	191	2	,	,	PUNCT
ejpam-5887	191	3	|vf⋆(0)−	|vf⋆(0)−	NOUN
ejpam-5887	191	4	vf⋆(j)|	vf⋆(j)|	X
ejpam-5887	191	5	>	>	SYM
ejpam-5887	191	6	1	1	NUM
ejpam-5887	191	7	for	for	ADP
ejpam-5887	191	8	some	some	PRON
ejpam-5887	191	9	j	j	NOUN
ejpam-5887	191	10	=	=	SYM
ejpam-5887	191	11	1	1	NUM
ejpam-5887	191	12	,	,	PUNCT
ejpam-5887	191	13	2	2	NUM
ejpam-5887	191	14	,	,	PUNCT
ejpam-5887	191	15	...	...	PUNCT
ejpam-5887	191	16	,	,	PUNCT
ejpam-5887	192	1	k	k	PROPN
ejpam-5887	192	2	−	−	PROPN
ejpam-5887	193	1	1	1	NUM
ejpam-5887	193	2	,	,	PUNCT
ejpam-5887	193	3	which	which	PRON
ejpam-5887	193	4	is	be	AUX
ejpam-5887	193	5	a	a	DET
ejpam-5887	193	6	contradiction	contradiction	NOUN
ejpam-5887	193	7	.	.	PUNCT
ejpam-5887	194	1	hence	hence	ADV
ejpam-5887	194	2	,	,	PUNCT
ejpam-5887	194	3	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	194	4	)	)	PUNCT
ejpam-5887	194	5	is	be	AUX
ejpam-5887	194	6	not	not	PART
ejpam-5887	194	7	an	an	DET
ejpam-5887	194	8	edge	edge	NOUN
ejpam-5887	194	9	k	k	NOUN
ejpam-5887	194	10	-	-	PUNCT
ejpam-5887	194	11	product	product	NOUN
ejpam-5887	194	12	cordial	cordial	ADJ
ejpam-5887	194	13	graph	graph	NOUN
ejpam-5887	194	14	for	for	ADP
ejpam-5887	194	15	k	k	PROPN
ejpam-5887	194	16	≤	≤	PROPN
ejpam-5887	194	17	n.	n.	PROPN
ejpam-5887	194	18	n.	n.	PROPN
ejpam-5887	194	19	m.	m.	NOUN
ejpam-5887	194	20	noureldeen	noureldeen	NOUN
ejpam-5887	194	21	et	et	PROPN
ejpam-5887	194	22	al	al	PROPN
ejpam-5887	194	23	.	.	PUNCT
ejpam-5887	194	24	/	/	SYM
ejpam-5887	194	25	eur	eur	PROPN
ejpam-5887	194	26	.	.	PUNCT
ejpam-5887	195	1	j.	j.	PROPN
ejpam-5887	195	2	pure	pure	PROPN
ejpam-5887	195	3	appl	appl	PROPN
ejpam-5887	195	4	.	.	PROPN
ejpam-5887	195	5	math	math	PROPN
ejpam-5887	195	6	,	,	PUNCT
ejpam-5887	195	7	18	18	NUM
ejpam-5887	195	8	(	(	PUNCT
ejpam-5887	195	9	2	2	NUM
ejpam-5887	195	10	)	)	PUNCT
ejpam-5887	195	11	(	(	PUNCT
ejpam-5887	195	12	2025	2025	NUM
ejpam-5887	195	13	)	)	PUNCT
ejpam-5887	195	14	,	,	PUNCT
ejpam-5887	195	15	5887	5887	NUM
ejpam-5887	195	16	7	7	NUM
ejpam-5887	195	17	of	of	ADP
ejpam-5887	195	18	21	21	NUM
ejpam-5887	195	19	theorem	theorem	NOUN
ejpam-5887	195	20	7	7	NUM
ejpam-5887	195	21	.	.	PUNCT
ejpam-5887	196	1	the	the	DET
ejpam-5887	196	2	graph	graph	NOUN
ejpam-5887	196	3	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	196	4	)	)	PUNCT
ejpam-5887	196	5	admits	admit	VERB
ejpam-5887	196	6	an	an	DET
ejpam-5887	196	7	edge	edge	NOUN
ejpam-5887	196	8	3	3	NUM
ejpam-5887	196	9	-	-	PUNCT
ejpam-5887	196	10	product	product	NOUN
ejpam-5887	196	11	cordial	cordial	ADJ
ejpam-5887	196	12	labeling	labeling	NOUN
ejpam-5887	196	13	if	if	SCONJ
ejpam-5887	196	14	and	and	CCONJ
ejpam-5887	196	15	only	only	ADV
ejpam-5887	196	16	if	if	SCONJ
ejpam-5887	196	17	n	n	PROPN
ejpam-5887	196	18	=	=	SYM
ejpam-5887	196	19	1	1	X
ejpam-5887	196	20	.	.	PUNCT
ejpam-5887	197	1	proof	proof	NOUN
ejpam-5887	197	2	.	.	PUNCT
ejpam-5887	198	1	let	let	VERB
ejpam-5887	198	2	the	the	DET
ejpam-5887	198	3	vertex	vertex	NOUN
ejpam-5887	198	4	set	set	NOUN
ejpam-5887	198	5	and	and	CCONJ
ejpam-5887	198	6	edge	edge	NOUN
ejpam-5887	198	7	set	set	NOUN
ejpam-5887	198	8	of	of	ADP
ejpam-5887	198	9	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	198	10	)	)	PUNCT
ejpam-5887	198	11	be	be	VERB
ejpam-5887	198	12	v	v	ADP
ejpam-5887	198	13	(	(	PUNCT
ejpam-5887	198	14	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	198	15	)	)	PUNCT
ejpam-5887	198	16	)	)	PUNCT
ejpam-5887	199	1	=	=	PRON
ejpam-5887	199	2	{	{	PUNCT
ejpam-5887	199	3	u	u	NOUN
ejpam-5887	199	4	,	,	PUNCT
ejpam-5887	199	5	v	v	NOUN
ejpam-5887	199	6	,	,	PUNCT
ejpam-5887	199	7	ui	ui	NOUN
ejpam-5887	199	8	,	,	PUNCT
ejpam-5887	199	9	vi	vi	PROPN
ejpam-5887	199	10	;	;	PUNCT
ejpam-5887	199	11	1	1	NUM
ejpam-5887	199	12	≤	≤	NUM
ejpam-5887	199	13	i	i	PRON
ejpam-5887	199	14	≤	≤	NOUN
ejpam-5887	199	15	n	n	CCONJ
ejpam-5887	199	16	}	}	PUNCT
ejpam-5887	199	17	and	and	CCONJ
ejpam-5887	199	18	e(d2(k1,n	e(d2(k1,n	NOUN
ejpam-5887	199	19	)	)	PUNCT
ejpam-5887	199	20	)	)	PUNCT
ejpam-5887	200	1	=	=	PRON
ejpam-5887	200	2	{	{	PUNCT
ejpam-5887	200	3	uui	uui	NOUN
ejpam-5887	200	4	,	,	PUNCT
ejpam-5887	200	5	vvi	vvi	PROPN
ejpam-5887	200	6	,	,	PUNCT
ejpam-5887	200	7	uvi	uvi	PROPN
ejpam-5887	200	8	,	,	PUNCT
ejpam-5887	200	9	vui	vui	PROPN
ejpam-5887	200	10	;	;	PUNCT
ejpam-5887	200	11	1	1	NUM
ejpam-5887	200	12	≤	≤	NUM
ejpam-5887	200	13	i	i	PRON
ejpam-5887	200	14	≤	≤	NOUN
ejpam-5887	200	15	n	n	CCONJ
ejpam-5887	200	16	}	}	PUNCT
ejpam-5887	200	17	respectively	respectively	ADV
ejpam-5887	200	18	.	.	PUNCT
ejpam-5887	200	19	define	define	VERB
ejpam-5887	200	20	the	the	DET
ejpam-5887	200	21	edge	edge	NOUN
ejpam-5887	200	22	labeling	labeling	NOUN
ejpam-5887	200	23	f	f	PROPN
ejpam-5887	200	24	:	:	PUNCT
ejpam-5887	200	25	e(d2(k1,1	e(d2(k1,1	NUM
ejpam-5887	200	26	)	)	PUNCT
ejpam-5887	200	27	)	)	PUNCT
ejpam-5887	201	1	→	→	PUNCT
ejpam-5887	201	2	{	{	PUNCT
ejpam-5887	201	3	0	0	NUM
ejpam-5887	201	4	,	,	PUNCT
ejpam-5887	201	5	1	1	NUM
ejpam-5887	201	6	,	,	PUNCT
ejpam-5887	201	7	2	2	NUM
ejpam-5887	201	8	}	}	PUNCT
ejpam-5887	201	9	as	as	SCONJ
ejpam-5887	201	10	follows	follow	VERB
ejpam-5887	201	11	:	:	PUNCT
ejpam-5887	201	12	f(uu1	f(uu1	NOUN
ejpam-5887	201	13	)	)	PUNCT
ejpam-5887	201	14	=	=	SYM
ejpam-5887	201	15	0	0	NUM
ejpam-5887	201	16	,	,	PUNCT
ejpam-5887	201	17	f(vv1	f(vv1	NOUN
ejpam-5887	201	18	)	)	PUNCT
ejpam-5887	201	19	=	=	PUNCT
ejpam-5887	201	20	f(uv1	f(uv1	X
ejpam-5887	201	21	)	)	PUNCT
ejpam-5887	201	22	=	=	SYM
ejpam-5887	201	23	1	1	NUM
ejpam-5887	201	24	,	,	PUNCT
ejpam-5887	201	25	f(vu1	f(vu1	NOUN
ejpam-5887	201	26	)	)	PUNCT
ejpam-5887	201	27	=	=	SYM
ejpam-5887	201	28	2	2	X
ejpam-5887	201	29	.	.	X
ejpam-5887	201	30	from	from	ADP
ejpam-5887	201	31	this	this	DET
ejpam-5887	201	32	labeling	labeling	NOUN
ejpam-5887	201	33	we	we	PRON
ejpam-5887	201	34	get	get	VERB
ejpam-5887	201	35	,	,	PUNCT
ejpam-5887	201	36	ef	ef	PROPN
ejpam-5887	201	37	(	(	PUNCT
ejpam-5887	201	38	0	0	NUM
ejpam-5887	201	39	)	)	PUNCT
ejpam-5887	201	40	=	=	SYM
ejpam-5887	201	41	ef	ef	X
ejpam-5887	201	42	(	(	PUNCT
ejpam-5887	201	43	1	1	NUM
ejpam-5887	201	44	)	)	PUNCT
ejpam-5887	201	45	−	−	PROPN
ejpam-5887	201	46	1	1	NUM
ejpam-5887	201	47	=	=	SYM
ejpam-5887	201	48	ef	ef	X
ejpam-5887	201	49	(	(	PUNCT
ejpam-5887	201	50	2	2	NUM
ejpam-5887	201	51	)	)	PUNCT
ejpam-5887	201	52	=	=	SYM
ejpam-5887	201	53	1	1	NUM
ejpam-5887	201	54	and	and	CCONJ
ejpam-5887	201	55	vf∗(0	vf∗(0	NOUN
ejpam-5887	201	56	)	)	PUNCT
ejpam-5887	201	57	−	−	PROPN
ejpam-5887	201	58	1	1	NUM
ejpam-5887	201	59	=	=	NOUN
ejpam-5887	201	60	vf∗(1	vf∗(1	X
ejpam-5887	201	61	)	)	PUNCT
ejpam-5887	201	62	=	=	SYM
ejpam-5887	201	63	vf∗(2	vf∗(2	ADJ
ejpam-5887	201	64	)	)	PUNCT
ejpam-5887	201	65	=	=	SYM
ejpam-5887	202	1	1	1	X
ejpam-5887	202	2	.	.	X
ejpam-5887	203	1	hence	hence	ADV
ejpam-5887	203	2	,	,	PUNCT
ejpam-5887	203	3	d2(k1,1	d2(k1,1	PROPN
ejpam-5887	203	4	)	)	PUNCT
ejpam-5887	203	5	is	be	AUX
ejpam-5887	203	6	an	an	DET
ejpam-5887	203	7	edge	edge	NOUN
ejpam-5887	203	8	3	3	NUM
ejpam-5887	203	9	-	-	PUNCT
ejpam-5887	203	10	product	product	NOUN
ejpam-5887	203	11	cordial	cordial	ADJ
ejpam-5887	203	12	graph	graph	NOUN
ejpam-5887	203	13	.	.	PUNCT
ejpam-5887	204	1	for	for	ADP
ejpam-5887	204	2	n	n	NOUN
ejpam-5887	204	3	=	=	SYM
ejpam-5887	204	4	2	2	NUM
ejpam-5887	204	5	,	,	PUNCT
ejpam-5887	204	6	|v	|v	ADV
ejpam-5887	204	7	|	|	NOUN
ejpam-5887	204	8	=	=	SYM
ejpam-5887	204	9	6	6	NUM
ejpam-5887	204	10	and	and	CCONJ
ejpam-5887	204	11	|e|	|e|	NOUN
ejpam-5887	204	12	=	=	ADJ
ejpam-5887	204	13	8	8	X
ejpam-5887	204	14	.	.	PUNCT
ejpam-5887	204	15	by	by	ADP
ejpam-5887	204	16	theorem	theorem	NOUN
ejpam-5887	204	17	5	5	NUM
ejpam-5887	204	18	,	,	PUNCT
ejpam-5887	204	19	d2(k1,2	d2(k1,2	ADJ
ejpam-5887	204	20	)	)	PUNCT
ejpam-5887	204	21	is	be	AUX
ejpam-5887	204	22	not	not	PART
ejpam-5887	204	23	an	an	DET
ejpam-5887	204	24	edge	edge	NOUN
ejpam-5887	204	25	3	3	NUM
ejpam-5887	204	26	-	-	PUNCT
ejpam-5887	204	27	product	product	NOUN
ejpam-5887	204	28	cordial	cordial	ADJ
ejpam-5887	204	29	graph	graph	NOUN
ejpam-5887	204	30	.	.	PUNCT
ejpam-5887	205	1	also	also	ADV
ejpam-5887	205	2	,	,	PUNCT
ejpam-5887	205	3	by	by	ADP
ejpam-5887	205	4	theorem	theorem	ADJ
ejpam-5887	205	5	6	6	NUM
ejpam-5887	205	6	,	,	PUNCT
ejpam-5887	205	7	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	205	8	)	)	PUNCT
ejpam-5887	205	9	;	;	PUNCT
ejpam-5887	205	10	n	n	PRON
ejpam-5887	205	11	≥	≥	NOUN
ejpam-5887	205	12	3	3	NUM
ejpam-5887	205	13	is	be	AUX
ejpam-5887	205	14	not	not	PART
ejpam-5887	205	15	an	an	DET
ejpam-5887	205	16	edge	edge	NOUN
ejpam-5887	205	17	3	3	NUM
ejpam-5887	205	18	-	-	PUNCT
ejpam-5887	205	19	product	product	NOUN
ejpam-5887	205	20	cordial	cordial	ADJ
ejpam-5887	205	21	graph	graph	NOUN
ejpam-5887	205	22	.	.	PUNCT
ejpam-5887	206	1	theorem	theorem	ADJ
ejpam-5887	206	2	8	8	NUM
ejpam-5887	206	3	.	.	PUNCT
ejpam-5887	207	1	the	the	DET
ejpam-5887	207	2	graph	graph	NOUN
ejpam-5887	207	3	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	207	4	)	)	PUNCT
ejpam-5887	207	5	does	do	AUX
ejpam-5887	207	6	not	not	PART
ejpam-5887	207	7	admit	admit	VERB
ejpam-5887	207	8	an	an	DET
ejpam-5887	207	9	edge	edge	NOUN
ejpam-5887	207	10	4	4	NUM
ejpam-5887	207	11	-	-	PUNCT
ejpam-5887	207	12	product	product	NOUN
ejpam-5887	207	13	cordial	cordial	ADJ
ejpam-5887	207	14	labeling	labeling	NOUN
ejpam-5887	207	15	.	.	PUNCT
ejpam-5887	208	1	proof	proof	NOUN
ejpam-5887	208	2	.	.	PUNCT
ejpam-5887	209	1	let	let	VERB
ejpam-5887	209	2	the	the	DET
ejpam-5887	209	3	vertex	vertex	NOUN
ejpam-5887	209	4	set	set	NOUN
ejpam-5887	209	5	and	and	CCONJ
ejpam-5887	209	6	edge	edge	NOUN
ejpam-5887	209	7	set	set	NOUN
ejpam-5887	209	8	of	of	ADP
ejpam-5887	209	9	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	209	10	)	)	PUNCT
ejpam-5887	209	11	be	be	VERB
ejpam-5887	209	12	v	v	ADP
ejpam-5887	209	13	(	(	PUNCT
ejpam-5887	209	14	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	209	15	)	)	PUNCT
ejpam-5887	209	16	)	)	PUNCT
ejpam-5887	210	1	=	=	PRON
ejpam-5887	210	2	{	{	PUNCT
ejpam-5887	210	3	u	u	NOUN
ejpam-5887	210	4	,	,	PUNCT
ejpam-5887	210	5	v	v	NOUN
ejpam-5887	210	6	,	,	PUNCT
ejpam-5887	210	7	ui	ui	NOUN
ejpam-5887	210	8	,	,	PUNCT
ejpam-5887	210	9	vi	vi	PROPN
ejpam-5887	210	10	;	;	PUNCT
ejpam-5887	210	11	1	1	NUM
ejpam-5887	210	12	≤	≤	NUM
ejpam-5887	210	13	i	i	PRON
ejpam-5887	210	14	≤	≤	NOUN
ejpam-5887	210	15	n	n	CCONJ
ejpam-5887	210	16	}	}	PUNCT
ejpam-5887	210	17	and	and	CCONJ
ejpam-5887	210	18	e(d2(k1,n	e(d2(k1,n	NOUN
ejpam-5887	210	19	)	)	PUNCT
ejpam-5887	210	20	)	)	PUNCT
ejpam-5887	211	1	=	=	PRON
ejpam-5887	211	2	{	{	PUNCT
ejpam-5887	211	3	uui	uui	NOUN
ejpam-5887	211	4	,	,	PUNCT
ejpam-5887	211	5	vvi	vvi	PROPN
ejpam-5887	211	6	,	,	PUNCT
ejpam-5887	211	7	uvi	uvi	PROPN
ejpam-5887	211	8	,	,	PUNCT
ejpam-5887	211	9	vui	vui	PROPN
ejpam-5887	211	10	;	;	PUNCT
ejpam-5887	211	11	1	1	NUM
ejpam-5887	211	12	≤	≤	NUM
ejpam-5887	211	13	i	i	PRON
ejpam-5887	211	14	≤	≤	NOUN
ejpam-5887	211	15	n	n	CCONJ
ejpam-5887	211	16	}	}	PUNCT
ejpam-5887	211	17	respectively	respectively	ADV
ejpam-5887	211	18	.	.	PUNCT
ejpam-5887	212	1	for	for	ADP
ejpam-5887	212	2	n	n	NOUN
ejpam-5887	212	3	=	=	SYM
ejpam-5887	212	4	1	1	NUM
ejpam-5887	212	5	,	,	PUNCT
ejpam-5887	212	6	|v	|v	ADV
ejpam-5887	212	7	|	|	NOUN
ejpam-5887	212	8	=	=	SYM
ejpam-5887	212	9	4	4	NUM
ejpam-5887	212	10	and	and	CCONJ
ejpam-5887	212	11	|e|	|e|	NOUN
ejpam-5887	212	12	=	=	SYM
ejpam-5887	212	13	4	4	NUM
ejpam-5887	212	14	.	.	NOUN
ejpam-5887	212	15	for	for	ADP
ejpam-5887	212	16	n	n	NOUN
ejpam-5887	212	17	=	=	SYM
ejpam-5887	212	18	3	3	NUM
ejpam-5887	212	19	,	,	PUNCT
ejpam-5887	212	20	|v	|v	ADV
ejpam-5887	212	21	|	|	NOUN
ejpam-5887	212	22	=	=	SYM
ejpam-5887	212	23	8	8	NUM
ejpam-5887	212	24	and	and	CCONJ
ejpam-5887	212	25	|e|	|e|	NOUN
ejpam-5887	212	26	=	=	NOUN
ejpam-5887	212	27	12	12	NUM
ejpam-5887	212	28	.	.	PUNCT
ejpam-5887	213	1	by	by	ADP
ejpam-5887	213	2	theorem	theorem	NOUN
ejpam-5887	213	3	5	5	NUM
ejpam-5887	213	4	,	,	PUNCT
ejpam-5887	213	5	d2(k1,1	d2(k1,1	PROPN
ejpam-5887	213	6	)	)	PUNCT
ejpam-5887	213	7	and	and	CCONJ
ejpam-5887	213	8	d2(k1,3	d2(k1,3	NOUN
ejpam-5887	213	9	)	)	PUNCT
ejpam-5887	213	10	are	be	AUX
ejpam-5887	213	11	not	not	PART
ejpam-5887	213	12	edge	edge	VERB
ejpam-5887	213	13	4	4	NUM
ejpam-5887	213	14	-	-	PUNCT
ejpam-5887	213	15	product	product	NOUN
ejpam-5887	213	16	cordial	cordial	ADJ
ejpam-5887	213	17	graphs	graph	NOUN
ejpam-5887	213	18	.	.	PUNCT
ejpam-5887	214	1	let	let	VERB
ejpam-5887	214	2	f	f	PRON
ejpam-5887	214	3	be	be	AUX
ejpam-5887	214	4	an	an	DET
ejpam-5887	214	5	edge	edge	NOUN
ejpam-5887	214	6	4	4	NUM
ejpam-5887	214	7	-	-	PUNCT
ejpam-5887	214	8	product	product	NOUN
ejpam-5887	214	9	cordial	cordial	ADJ
ejpam-5887	214	10	labeling	labeling	NOUN
ejpam-5887	214	11	of	of	ADP
ejpam-5887	214	12	d2(k1,2	d2(k1,2	NOUN
ejpam-5887	214	13	)	)	PUNCT
ejpam-5887	214	14	.	.	PUNCT
ejpam-5887	215	1	then	then	ADV
ejpam-5887	215	2	,	,	PUNCT
ejpam-5887	215	3	ef	ef	PROPN
ejpam-5887	215	4	(	(	PUNCT
ejpam-5887	215	5	i	i	NOUN
ejpam-5887	215	6	)	)	PUNCT
ejpam-5887	215	7	=	=	SYM
ejpam-5887	215	8	2	2	NUM
ejpam-5887	215	9	and	and	CCONJ
ejpam-5887	215	10	vf∗(i	vf∗(i	ADJ
ejpam-5887	215	11	)	)	PUNCT
ejpam-5887	215	12	=	=	SYM
ejpam-5887	215	13	1	1	NUM
ejpam-5887	215	14	or	or	CCONJ
ejpam-5887	215	15	2	2	NUM
ejpam-5887	215	16	.	.	PUNCT
ejpam-5887	216	1	if	if	SCONJ
ejpam-5887	216	2	ef	ef	PROPN
ejpam-5887	216	3	(	(	PUNCT
ejpam-5887	216	4	0	0	NUM
ejpam-5887	216	5	)	)	PUNCT
ejpam-5887	216	6	=	=	SYM
ejpam-5887	216	7	2	2	NUM
ejpam-5887	216	8	,	,	PUNCT
ejpam-5887	216	9	then	then	ADV
ejpam-5887	216	10	vf∗(0	vf∗(0	VERB
ejpam-5887	216	11	)	)	PUNCT
ejpam-5887	216	12	≥	≥	NOUN
ejpam-5887	216	13	3	3	NUM
ejpam-5887	216	14	.	.	PUNCT
ejpam-5887	216	15	therefore	therefore	ADV
ejpam-5887	216	16	,	,	PUNCT
ejpam-5887	216	17	|vf∗(0	|vf∗(0	X
ejpam-5887	216	18	)	)	PUNCT
ejpam-5887	216	19	−	−	NOUN
ejpam-5887	216	20	vf∗(j)|	vf∗(j)|	NOUN
ejpam-5887	216	21	>	>	X
ejpam-5887	216	22	1	1	NUM
ejpam-5887	216	23	for	for	ADP
ejpam-5887	216	24	some	some	PRON
ejpam-5887	216	25	j	j	NOUN
ejpam-5887	216	26	=	=	SYM
ejpam-5887	216	27	1	1	NUM
ejpam-5887	216	28	,	,	PUNCT
ejpam-5887	216	29	2	2	NUM
ejpam-5887	216	30	,	,	PUNCT
ejpam-5887	216	31	3	3	NUM
ejpam-5887	216	32	,	,	PUNCT
ejpam-5887	216	33	which	which	PRON
ejpam-5887	216	34	is	be	AUX
ejpam-5887	216	35	a	a	DET
ejpam-5887	216	36	contradiction	contradiction	NOUN
ejpam-5887	216	37	.	.	PUNCT
ejpam-5887	217	1	hence	hence	ADV
ejpam-5887	217	2	,	,	PUNCT
ejpam-5887	217	3	d2(k1,2	d2(k1,2	ADJ
ejpam-5887	217	4	)	)	PUNCT
ejpam-5887	217	5	is	be	AUX
ejpam-5887	217	6	not	not	PART
ejpam-5887	217	7	an	an	DET
ejpam-5887	217	8	edge	edge	NOUN
ejpam-5887	217	9	4	4	NUM
ejpam-5887	217	10	-	-	PUNCT
ejpam-5887	217	11	product	product	NOUN
ejpam-5887	217	12	cordial	cordial	ADJ
ejpam-5887	217	13	graph	graph	NOUN
ejpam-5887	217	14	.	.	PUNCT
ejpam-5887	218	1	clearly	clearly	ADV
ejpam-5887	218	2	,	,	PUNCT
ejpam-5887	218	3	by	by	ADP
ejpam-5887	218	4	theorem	theorem	ADJ
ejpam-5887	218	5	6	6	NUM
ejpam-5887	218	6	,	,	PUNCT
ejpam-5887	218	7	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	218	8	)	)	PUNCT
ejpam-5887	218	9	is	be	AUX
ejpam-5887	218	10	not	not	PART
ejpam-5887	218	11	an	an	DET
ejpam-5887	218	12	edge	edge	NOUN
ejpam-5887	218	13	4	4	NUM
ejpam-5887	218	14	-	-	PUNCT
ejpam-5887	218	15	product	product	NOUN
ejpam-5887	218	16	cordial	cordial	ADJ
ejpam-5887	218	17	graph	graph	NOUN
ejpam-5887	218	18	if	if	SCONJ
ejpam-5887	218	19	n	n	NUM
ejpam-5887	218	20	≥	≥	NOUN
ejpam-5887	218	21	4	4	NUM
ejpam-5887	218	22	.	.	PUNCT
ejpam-5887	218	23	theorem	theorem	VERB
ejpam-5887	218	24	9	9	NUM
ejpam-5887	218	25	.	.	PUNCT
ejpam-5887	219	1	the	the	DET
ejpam-5887	219	2	graph	graph	NOUN
ejpam-5887	219	3	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	219	4	)	)	PUNCT
ejpam-5887	219	5	admits	admit	VERB
ejpam-5887	219	6	an	an	DET
ejpam-5887	219	7	edge	edge	NOUN
ejpam-5887	219	8	5	5	NUM
ejpam-5887	219	9	-	-	PUNCT
ejpam-5887	219	10	product	product	NOUN
ejpam-5887	219	11	cordial	cordial	ADJ
ejpam-5887	219	12	labeling	labeling	NOUN
ejpam-5887	219	13	if	if	SCONJ
ejpam-5887	219	14	and	and	CCONJ
ejpam-5887	219	15	only	only	ADV
ejpam-5887	219	16	if	if	SCONJ
ejpam-5887	219	17	n	n	NOUN
ejpam-5887	219	18	=	=	SYM
ejpam-5887	219	19	2	2	X
ejpam-5887	219	20	.	.	PUNCT
ejpam-5887	220	1	proof	proof	NOUN
ejpam-5887	220	2	.	.	PUNCT
ejpam-5887	221	1	let	let	VERB
ejpam-5887	221	2	the	the	DET
ejpam-5887	221	3	vertex	vertex	NOUN
ejpam-5887	221	4	set	set	NOUN
ejpam-5887	221	5	and	and	CCONJ
ejpam-5887	221	6	edge	edge	NOUN
ejpam-5887	221	7	set	set	NOUN
ejpam-5887	221	8	of	of	ADP
ejpam-5887	221	9	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	221	10	)	)	PUNCT
ejpam-5887	221	11	be	be	VERB
ejpam-5887	221	12	v	v	ADP
ejpam-5887	221	13	(	(	PUNCT
ejpam-5887	221	14	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	221	15	)	)	PUNCT
ejpam-5887	221	16	)	)	PUNCT
ejpam-5887	222	1	=	=	PRON
ejpam-5887	222	2	{	{	PUNCT
ejpam-5887	222	3	u	u	NOUN
ejpam-5887	222	4	,	,	PUNCT
ejpam-5887	222	5	v	v	NOUN
ejpam-5887	222	6	,	,	PUNCT
ejpam-5887	222	7	ui	ui	NOUN
ejpam-5887	222	8	,	,	PUNCT
ejpam-5887	222	9	vi	vi	PROPN
ejpam-5887	222	10	;	;	PUNCT
ejpam-5887	222	11	1	1	NUM
ejpam-5887	222	12	≤	≤	NUM
ejpam-5887	222	13	i	i	PRON
ejpam-5887	222	14	≤	≤	NOUN
ejpam-5887	222	15	n	n	CCONJ
ejpam-5887	222	16	}	}	PUNCT
ejpam-5887	222	17	and	and	CCONJ
ejpam-5887	222	18	e(d2(k1,n	e(d2(k1,n	NOUN
ejpam-5887	222	19	)	)	PUNCT
ejpam-5887	222	20	)	)	PUNCT
ejpam-5887	223	1	=	=	PRON
ejpam-5887	223	2	{	{	PUNCT
ejpam-5887	223	3	uui	uui	NOUN
ejpam-5887	223	4	,	,	PUNCT
ejpam-5887	223	5	vvi	vvi	PROPN
ejpam-5887	223	6	,	,	PUNCT
ejpam-5887	223	7	uvi	uvi	PROPN
ejpam-5887	223	8	,	,	PUNCT
ejpam-5887	223	9	vui	vui	PROPN
ejpam-5887	223	10	;	;	PUNCT
ejpam-5887	223	11	1	1	NUM
ejpam-5887	223	12	≤	≤	NUM
ejpam-5887	223	13	i	i	PRON
ejpam-5887	223	14	≤	≤	NOUN
ejpam-5887	223	15	n	n	CCONJ
ejpam-5887	223	16	}	}	PUNCT
ejpam-5887	223	17	respectively	respectively	ADV
ejpam-5887	223	18	.	.	PUNCT
ejpam-5887	223	19	define	define	VERB
ejpam-5887	223	20	the	the	DET
ejpam-5887	223	21	edge	edge	NOUN
ejpam-5887	223	22	labeling	labeling	NOUN
ejpam-5887	223	23	f	f	NOUN
ejpam-5887	223	24	:	:	PUNCT
ejpam-5887	223	25	e(d2(k1,2	e(d2(k1,2	NUM
ejpam-5887	223	26	)	)	PUNCT
ejpam-5887	223	27	)	)	PUNCT
ejpam-5887	223	28	→	→	PUNCT
ejpam-5887	223	29	{	{	PUNCT
ejpam-5887	223	30	0	0	NUM
ejpam-5887	223	31	,	,	PUNCT
ejpam-5887	223	32	1	1	NUM
ejpam-5887	223	33	,	,	PUNCT
ejpam-5887	223	34	2	2	NUM
ejpam-5887	223	35	,	,	PUNCT
ejpam-5887	223	36	3	3	NUM
ejpam-5887	223	37	,	,	PUNCT
ejpam-5887	223	38	4	4	NUM
ejpam-5887	223	39	}	}	PUNCT
ejpam-5887	223	40	as	as	ADP
ejpam-5887	223	41	f(uu1	f(uu1	PROPN
ejpam-5887	223	42	)	)	PUNCT
ejpam-5887	223	43	=	=	SYM
ejpam-5887	223	44	4	4	NUM
ejpam-5887	223	45	,	,	PUNCT
ejpam-5887	223	46	f(uu2	f(uu2	ADJ
ejpam-5887	223	47	)	)	PUNCT
ejpam-5887	223	48	=	=	SYM
ejpam-5887	223	49	1	1	NUM
ejpam-5887	223	50	,	,	PUNCT
ejpam-5887	223	51	f(uv1	f(uv1	NOUN
ejpam-5887	223	52	)	)	PUNCT
ejpam-5887	223	53	=	=	SYM
ejpam-5887	223	54	3	3	NUM
ejpam-5887	223	55	,	,	PUNCT
ejpam-5887	223	56	f(uv2	f(uv2	ADJ
ejpam-5887	223	57	)	)	PUNCT
ejpam-5887	223	58	=	=	SYM
ejpam-5887	223	59	0	0	NUM
ejpam-5887	223	60	,	,	PUNCT
ejpam-5887	223	61	f(vu1	f(vu1	NOUN
ejpam-5887	223	62	)	)	PUNCT
ejpam-5887	223	63	=	=	SYM
ejpam-5887	223	64	2	2	NUM
ejpam-5887	223	65	,	,	PUNCT
ejpam-5887	223	66	f(vu2	f(vu2	NOUN
ejpam-5887	223	67	)	)	PUNCT
ejpam-5887	223	68	=	=	SYM
ejpam-5887	223	69	1	1	NUM
ejpam-5887	223	70	,	,	PUNCT
ejpam-5887	223	71	f(vv1	f(vv1	NOUN
ejpam-5887	223	72	)	)	PUNCT
ejpam-5887	223	73	=	=	SYM
ejpam-5887	223	74	4	4	NUM
ejpam-5887	223	75	,	,	PUNCT
ejpam-5887	223	76	f(vv2	f(vv2	PROPN
ejpam-5887	223	77	)	)	PUNCT
ejpam-5887	223	78	=	=	PUNCT
ejpam-5887	224	1	3	3	X
ejpam-5887	224	2	.	.	X
ejpam-5887	224	3	from	from	ADP
ejpam-5887	224	4	this	this	DET
ejpam-5887	224	5	labeling	labeling	NOUN
ejpam-5887	224	6	we	we	PRON
ejpam-5887	224	7	get	get	VERB
ejpam-5887	224	8	,	,	PUNCT
ejpam-5887	224	9	ef	ef	PROPN
ejpam-5887	224	10	(	(	PUNCT
ejpam-5887	224	11	0	0	NUM
ejpam-5887	224	12	)	)	PUNCT
ejpam-5887	224	13	+	+	CCONJ
ejpam-5887	224	14	1	1	X
ejpam-5887	224	15	=	=	SYM
ejpam-5887	224	16	ef	ef	X
ejpam-5887	224	17	(	(	PUNCT
ejpam-5887	224	18	1	1	NUM
ejpam-5887	224	19	)	)	PUNCT
ejpam-5887	224	20	=	=	SYM
ejpam-5887	224	21	ef	ef	X
ejpam-5887	224	22	(	(	PUNCT
ejpam-5887	224	23	2	2	NUM
ejpam-5887	224	24	)	)	PUNCT
ejpam-5887	224	25	+	+	CCONJ
ejpam-5887	224	26	1	1	NUM
ejpam-5887	224	27	=	=	SYM
ejpam-5887	224	28	ef	ef	X
ejpam-5887	224	29	(	(	PUNCT
ejpam-5887	224	30	3	3	NUM
ejpam-5887	224	31	)	)	PUNCT
ejpam-5887	224	32	=	=	SYM
ejpam-5887	224	33	ef	ef	X
ejpam-5887	224	34	(	(	PUNCT
ejpam-5887	224	35	4	4	NUM
ejpam-5887	224	36	)	)	PUNCT
ejpam-5887	224	37	=	=	SYM
ejpam-5887	224	38	2	2	NUM
ejpam-5887	224	39	and	and	CCONJ
ejpam-5887	224	40	vf∗(0)−	vf∗(0)−	NOUN
ejpam-5887	224	41	1	1	NUM
ejpam-5887	224	42	=	=	NOUN
ejpam-5887	224	43	vf∗(1	vf∗(1	X
ejpam-5887	224	44	)	)	PUNCT
ejpam-5887	224	45	=	=	SYM
ejpam-5887	224	46	vf∗(2	vf∗(2	ADJ
ejpam-5887	224	47	)	)	PUNCT
ejpam-5887	224	48	=	=	SYM
ejpam-5887	224	49	v∗(3	v∗(3	NOUN
ejpam-5887	224	50	)	)	PUNCT
ejpam-5887	224	51	=	=	PUNCT
ejpam-5887	224	52	vf∗(4	vf∗(4	NOUN
ejpam-5887	224	53	)	)	PUNCT
ejpam-5887	224	54	=	=	SYM
ejpam-5887	225	1	1	1	X
ejpam-5887	225	2	.	.	PUNCT
ejpam-5887	225	3	hence	hence	ADV
ejpam-5887	225	4	,	,	PUNCT
ejpam-5887	225	5	d2(k1,2	d2(k1,2	PROPN
ejpam-5887	225	6	)	)	PUNCT
ejpam-5887	225	7	is	be	AUX
ejpam-5887	225	8	an	an	DET
ejpam-5887	225	9	edge	edge	NOUN
ejpam-5887	225	10	5	5	NUM
ejpam-5887	225	11	-	-	PUNCT
ejpam-5887	225	12	product	product	NOUN
ejpam-5887	225	13	cordial	cordial	ADJ
ejpam-5887	225	14	graph	graph	NOUN
ejpam-5887	225	15	.	.	PUNCT
ejpam-5887	226	1	for	for	ADP
ejpam-5887	226	2	n	n	NOUN
ejpam-5887	226	3	=	=	SYM
ejpam-5887	226	4	1	1	NUM
ejpam-5887	226	5	,	,	PUNCT
ejpam-5887	226	6	|v	|v	ADV
ejpam-5887	226	7	|	|	NOUN
ejpam-5887	227	1	=	=	SYM
ejpam-5887	227	2	|e|	|e|	NOUN
ejpam-5887	227	3	=	=	SYM
ejpam-5887	227	4	4	4	X
ejpam-5887	227	5	.	.	PUNCT
ejpam-5887	228	1	if	if	SCONJ
ejpam-5887	228	2	f	f	PROPN
ejpam-5887	228	3	is	be	AUX
ejpam-5887	228	4	an	an	DET
ejpam-5887	228	5	edge	edge	NOUN
ejpam-5887	228	6	5	5	NUM
ejpam-5887	228	7	-	-	PUNCT
ejpam-5887	228	8	product	product	NOUN
ejpam-5887	228	9	cordial	cordial	ADJ
ejpam-5887	228	10	labeling	labeling	NOUN
ejpam-5887	228	11	of	of	ADP
ejpam-5887	228	12	d2(k1,1	d2(k1,1	NOUN
ejpam-5887	228	13	)	)	PUNCT
ejpam-5887	228	14	,	,	PUNCT
ejpam-5887	228	15	then	then	ADV
ejpam-5887	228	16	ef	ef	PROPN
ejpam-5887	228	17	(	(	PUNCT
ejpam-5887	228	18	i	i	PROPN
ejpam-5887	228	19	)	)	PUNCT
ejpam-5887	228	20	and	and	CCONJ
ejpam-5887	228	21	vf∗(i	vf∗(i	ADJ
ejpam-5887	228	22	)	)	PUNCT
ejpam-5887	228	23	are	be	AUX
ejpam-5887	228	24	either	either	CCONJ
ejpam-5887	228	25	0	0	NUM
ejpam-5887	228	26	or	or	CCONJ
ejpam-5887	228	27	1	1	NUM
ejpam-5887	228	28	for	for	ADP
ejpam-5887	228	29	i	i	PRON
ejpam-5887	228	30	=	=	SYM
ejpam-5887	228	31	0	0	NUM
ejpam-5887	228	32	,	,	PUNCT
ejpam-5887	228	33	1	1	NUM
ejpam-5887	228	34	,	,	PUNCT
ejpam-5887	228	35	2	2	NUM
ejpam-5887	228	36	,	,	PUNCT
ejpam-5887	228	37	3	3	NUM
ejpam-5887	228	38	,	,	PUNCT
ejpam-5887	228	39	4	4	NUM
ejpam-5887	228	40	.	.	PUNCT
ejpam-5887	228	41	clearly	clearly	ADV
ejpam-5887	228	42	,	,	PUNCT
ejpam-5887	228	43	ef	ef	PROPN
ejpam-5887	228	44	(	(	PUNCT
ejpam-5887	228	45	0	0	NUM
ejpam-5887	228	46	)	)	PUNCT
ejpam-5887	228	47	=	=	SYM
ejpam-5887	228	48	0	0	NUM
ejpam-5887	228	49	otherwise	otherwise	ADV
ejpam-5887	228	50	vf∗(0	vf∗(0	NOUN
ejpam-5887	228	51	)	)	PUNCT
ejpam-5887	228	52	=	=	SYM
ejpam-5887	228	53	2	2	X
ejpam-5887	228	54	.	.	PUNCT
ejpam-5887	229	1	so	so	ADV
ejpam-5887	229	2	,	,	PUNCT
ejpam-5887	229	3	ef	ef	PROPN
ejpam-5887	229	4	(	(	PUNCT
ejpam-5887	229	5	i	i	NOUN
ejpam-5887	229	6	)	)	PUNCT
ejpam-5887	229	7	=	=	SYM
ejpam-5887	229	8	vf∗(i	vf∗(i	ADJ
ejpam-5887	229	9	)	)	PUNCT
ejpam-5887	229	10	=	=	SYM
ejpam-5887	229	11	1	1	X
ejpam-5887	229	12	(	(	PUNCT
ejpam-5887	229	13	i	i	NOUN
ejpam-5887	229	14	=	=	NOUN
ejpam-5887	229	15	1	1	NUM
ejpam-5887	229	16	,	,	PUNCT
ejpam-5887	229	17	2	2	NUM
ejpam-5887	229	18	,	,	PUNCT
ejpam-5887	229	19	3	3	NUM
ejpam-5887	229	20	,	,	PUNCT
ejpam-5887	229	21	4	4	NUM
ejpam-5887	229	22	)	)	PUNCT
ejpam-5887	229	23	.	.	PUNCT
ejpam-5887	230	1	in	in	ADP
ejpam-5887	230	2	order	order	NOUN
ejpam-5887	230	3	to	to	PART
ejpam-5887	230	4	get	get	VERB
ejpam-5887	230	5	the	the	DET
ejpam-5887	230	6	vertex	vertex	NOUN
ejpam-5887	230	7	label	label	NOUN
ejpam-5887	230	8	1	1	NUM
ejpam-5887	230	9	,	,	PUNCT
ejpam-5887	230	10	there	there	PRON
ejpam-5887	230	11	must	must	AUX
ejpam-5887	230	12	be	be	AUX
ejpam-5887	230	13	two	two	NUM
ejpam-5887	230	14	adjacent	adjacent	ADJ
ejpam-5887	230	15	edges	edge	NOUN
ejpam-5887	230	16	,	,	PUNCT
ejpam-5887	230	17	say	say	VERB
ejpam-5887	230	18	uu1	uu1	ADV
ejpam-5887	230	19	and	and	CCONJ
ejpam-5887	230	20	uv1	uv1	PROPN
ejpam-5887	230	21	with	with	ADP
ejpam-5887	230	22	labels	label	NOUN
ejpam-5887	230	23	2	2	NUM
ejpam-5887	230	24	and	and	CCONJ
ejpam-5887	230	25	3	3	NUM
ejpam-5887	230	26	respectively	respectively	ADV
ejpam-5887	230	27	.	.	PUNCT
ejpam-5887	231	1	to	to	PART
ejpam-5887	231	2	get	get	VERB
ejpam-5887	231	3	the	the	DET
ejpam-5887	231	4	vertex	vertex	NOUN
ejpam-5887	231	5	label	label	NOUN
ejpam-5887	231	6	2	2	NUM
ejpam-5887	231	7	,	,	PUNCT
ejpam-5887	231	8	we	we	PRON
ejpam-5887	231	9	must	must	AUX
ejpam-5887	231	10	have	have	VERB
ejpam-5887	231	11	f(vu1	f(vu1	NOUN
ejpam-5887	231	12	)	)	PUNCT
ejpam-5887	231	13	=	=	SYM
ejpam-5887	231	14	1	1	NUM
ejpam-5887	231	15	and	and	CCONJ
ejpam-5887	231	16	f(vv1	f(vv1	NOUN
ejpam-5887	231	17	)	)	PUNCT
ejpam-5887	231	18	=	=	SYM
ejpam-5887	231	19	4	4	NUM
ejpam-5887	231	20	,	,	PUNCT
ejpam-5887	231	21	which	which	PRON
ejpam-5887	231	22	results	result	VERB
ejpam-5887	231	23	in	in	ADP
ejpam-5887	231	24	vf∗(2	vf∗(2	ADJ
ejpam-5887	231	25	)	)	PUNCT
ejpam-5887	232	1	=	=	SYM
ejpam-5887	232	2	2	2	NUM
ejpam-5887	232	3	,	,	PUNCT
ejpam-5887	232	4	which	which	PRON
ejpam-5887	232	5	is	be	AUX
ejpam-5887	232	6	a	a	DET
ejpam-5887	232	7	contradiction	contradiction	NOUN
ejpam-5887	232	8	.	.	PUNCT
ejpam-5887	233	1	hence	hence	ADV
ejpam-5887	233	2	,	,	PUNCT
ejpam-5887	233	3	d2(k1,1	d2(k1,1	PROPN
ejpam-5887	233	4	)	)	PUNCT
ejpam-5887	233	5	is	be	AUX
ejpam-5887	233	6	not	not	PART
ejpam-5887	233	7	an	an	DET
ejpam-5887	233	8	edge	edge	NOUN
ejpam-5887	233	9	5	5	NUM
ejpam-5887	233	10	-	-	PUNCT
ejpam-5887	233	11	product	product	NOUN
ejpam-5887	233	12	cordial	cordial	ADJ
ejpam-5887	233	13	graph	graph	NOUN
ejpam-5887	233	14	.	.	PUNCT
ejpam-5887	234	1	for	for	ADP
ejpam-5887	234	2	n	n	NOUN
ejpam-5887	234	3	=	=	SYM
ejpam-5887	234	4	3	3	NUM
ejpam-5887	234	5	,	,	PUNCT
ejpam-5887	234	6	|v	|v	ADV
ejpam-5887	234	7	|	|	NOUN
ejpam-5887	234	8	=	=	SYM
ejpam-5887	234	9	8	8	NUM
ejpam-5887	234	10	and	and	CCONJ
ejpam-5887	234	11	|e|	|e|	NOUN
ejpam-5887	234	12	=	=	SYM
ejpam-5887	234	13	12	12	NUM
ejpam-5887	234	14	.	.	PUNCT
ejpam-5887	235	1	let	let	VERB
ejpam-5887	235	2	f	f	PRON
ejpam-5887	235	3	be	be	AUX
ejpam-5887	235	4	an	an	DET
ejpam-5887	235	5	edge	edge	NOUN
ejpam-5887	235	6	3	3	NUM
ejpam-5887	235	7	-	-	PUNCT
ejpam-5887	235	8	product	product	NOUN
ejpam-5887	235	9	cordial	cordial	ADJ
ejpam-5887	235	10	labeling	labeling	NOUN
ejpam-5887	235	11	of	of	ADP
ejpam-5887	235	12	d2(k1,3	d2(k1,3	NOUN
ejpam-5887	235	13	)	)	PUNCT
ejpam-5887	235	14	.	.	PUNCT
ejpam-5887	236	1	then	then	ADV
ejpam-5887	236	2	ef	ef	PROPN
ejpam-5887	236	3	(	(	PUNCT
ejpam-5887	236	4	i	i	NOUN
ejpam-5887	236	5	)	)	PUNCT
ejpam-5887	236	6	=	=	SYM
ejpam-5887	236	7	2	2	NUM
ejpam-5887	236	8	or	or	CCONJ
ejpam-5887	236	9	3	3	NUM
ejpam-5887	236	10	(	(	PUNCT
ejpam-5887	236	11	i	i	NOUN
ejpam-5887	236	12	=	=	NOUN
ejpam-5887	236	13	0	0	NUM
ejpam-5887	236	14	,	,	PUNCT
ejpam-5887	236	15	1	1	NUM
ejpam-5887	236	16	,	,	PUNCT
ejpam-5887	236	17	2	2	NUM
ejpam-5887	236	18	,	,	PUNCT
ejpam-5887	236	19	3	3	NUM
ejpam-5887	236	20	,	,	PUNCT
ejpam-5887	236	21	4	4	NUM
ejpam-5887	236	22	)	)	PUNCT
ejpam-5887	236	23	and	and	CCONJ
ejpam-5887	236	24	vf∗(i	vf∗(i	ADJ
ejpam-5887	236	25	)	)	PUNCT
ejpam-5887	236	26	=	=	SYM
ejpam-5887	236	27	1	1	NUM
ejpam-5887	236	28	or	or	CCONJ
ejpam-5887	236	29	2	2	NUM
ejpam-5887	236	30	(	(	PUNCT
ejpam-5887	236	31	i	i	NOUN
ejpam-5887	236	32	=	=	NOUN
ejpam-5887	236	33	0	0	NUM
ejpam-5887	236	34	,	,	PUNCT
ejpam-5887	236	35	1	1	NUM
ejpam-5887	236	36	,	,	PUNCT
ejpam-5887	236	37	2	2	NUM
ejpam-5887	236	38	,	,	PUNCT
ejpam-5887	236	39	3	3	NUM
ejpam-5887	236	40	,	,	PUNCT
ejpam-5887	236	41	4	4	NUM
ejpam-5887	236	42	)	)	PUNCT
ejpam-5887	236	43	.	.	PUNCT
ejpam-5887	237	1	now	now	ADV
ejpam-5887	237	2	,	,	PUNCT
ejpam-5887	237	3	ef	ef	PROPN
ejpam-5887	237	4	(	(	PUNCT
ejpam-5887	237	5	0	0	NUM
ejpam-5887	237	6	)	)	PUNCT
ejpam-5887	237	7	=	=	SYM
ejpam-5887	237	8	2	2	NUM
ejpam-5887	237	9	n.	n.	NOUN
ejpam-5887	237	10	m.	m.	NOUN
ejpam-5887	237	11	noureldeen	noureldeen	NOUN
ejpam-5887	237	12	et	et	PROPN
ejpam-5887	237	13	al	al	PROPN
ejpam-5887	237	14	.	.	PUNCT
ejpam-5887	237	15	/	/	SYM
ejpam-5887	237	16	eur	eur	PROPN
ejpam-5887	237	17	.	.	PUNCT
ejpam-5887	238	1	j.	j.	PROPN
ejpam-5887	238	2	pure	pure	PROPN
ejpam-5887	238	3	appl	appl	PROPN
ejpam-5887	238	4	.	.	PROPN
ejpam-5887	238	5	math	math	PROPN
ejpam-5887	238	6	,	,	PUNCT
ejpam-5887	238	7	18	18	NUM
ejpam-5887	238	8	(	(	PUNCT
ejpam-5887	238	9	2	2	NUM
ejpam-5887	238	10	)	)	PUNCT
ejpam-5887	238	11	(	(	PUNCT
ejpam-5887	238	12	2025	2025	NUM
ejpam-5887	238	13	)	)	PUNCT
ejpam-5887	238	14	,	,	PUNCT
ejpam-5887	238	15	5887	5887	NUM
ejpam-5887	238	16	8	8	NUM
ejpam-5887	238	17	of	of	ADP
ejpam-5887	238	18	21	21	NUM
ejpam-5887	238	19	implies	imply	VERB
ejpam-5887	238	20	vf∗(0	vf∗(0	NOUN
ejpam-5887	238	21	)	)	PUNCT
ejpam-5887	238	22	≥	≥	NOUN
ejpam-5887	238	23	3	3	NUM
ejpam-5887	238	24	>	>	SYM
ejpam-5887	238	25	2	2	NUM
ejpam-5887	238	26	.	.	PUNCT
ejpam-5887	239	1	therefore	therefore	ADV
ejpam-5887	239	2	|vf⋆(0)−	|vf⋆(0)−	PROPN
ejpam-5887	239	3	vf⋆(j)|	vf⋆(j)|	SYM
ejpam-5887	239	4	>	>	X
ejpam-5887	239	5	1	1	NUM
ejpam-5887	239	6	for	for	ADP
ejpam-5887	239	7	some	some	PRON
ejpam-5887	239	8	j	j	NOUN
ejpam-5887	239	9	=	=	SYM
ejpam-5887	239	10	1	1	NUM
ejpam-5887	239	11	,	,	PUNCT
ejpam-5887	239	12	2	2	NUM
ejpam-5887	239	13	,	,	PUNCT
ejpam-5887	239	14	3	3	NUM
ejpam-5887	239	15	,	,	PUNCT
ejpam-5887	239	16	4	4	NUM
ejpam-5887	239	17	,	,	PUNCT
ejpam-5887	239	18	which	which	PRON
ejpam-5887	239	19	is	be	AUX
ejpam-5887	239	20	a	a	DET
ejpam-5887	239	21	contradiction	contradiction	NOUN
ejpam-5887	239	22	.	.	PUNCT
ejpam-5887	240	1	hence	hence	ADV
ejpam-5887	240	2	,	,	PUNCT
ejpam-5887	240	3	d2(k1,3	d2(k1,3	PROPN
ejpam-5887	240	4	)	)	PUNCT
ejpam-5887	240	5	is	be	AUX
ejpam-5887	240	6	not	not	PART
ejpam-5887	240	7	an	an	DET
ejpam-5887	240	8	edge	edge	NOUN
ejpam-5887	240	9	5	5	NUM
ejpam-5887	240	10	-	-	PUNCT
ejpam-5887	240	11	product	product	NOUN
ejpam-5887	240	12	cordial	cordial	ADJ
ejpam-5887	240	13	graph	graph	NOUN
ejpam-5887	240	14	.	.	PUNCT
ejpam-5887	241	1	for	for	ADP
ejpam-5887	241	2	n	n	NOUN
ejpam-5887	241	3	=	=	SYM
ejpam-5887	241	4	4	4	NUM
ejpam-5887	241	5	,	,	PUNCT
ejpam-5887	241	6	|v	|v	ADV
ejpam-5887	241	7	|	|	NOUN
ejpam-5887	241	8	=	=	NOUN
ejpam-5887	241	9	10	10	NUM
ejpam-5887	241	10	and	and	CCONJ
ejpam-5887	241	11	|e|	|e|	ADJ
ejpam-5887	241	12	=	=	SYM
ejpam-5887	241	13	16	16	NUM
ejpam-5887	241	14	.	.	PUNCT
ejpam-5887	242	1	by	by	ADP
ejpam-5887	242	2	theorem	theorem	NOUN
ejpam-5887	242	3	5	5	NUM
ejpam-5887	242	4	,	,	PUNCT
ejpam-5887	242	5	d2(k1,4	d2(k1,4	NOUN
ejpam-5887	242	6	)	)	PUNCT
ejpam-5887	242	7	is	be	AUX
ejpam-5887	242	8	not	not	PART
ejpam-5887	242	9	an	an	DET
ejpam-5887	242	10	edge	edge	NOUN
ejpam-5887	242	11	5	5	NUM
ejpam-5887	242	12	-	-	PUNCT
ejpam-5887	242	13	product	product	NOUN
ejpam-5887	242	14	cordial	cordial	ADJ
ejpam-5887	242	15	graph	graph	NOUN
ejpam-5887	242	16	.	.	PUNCT
ejpam-5887	243	1	also	also	ADV
ejpam-5887	243	2	,	,	PUNCT
ejpam-5887	243	3	by	by	ADP
ejpam-5887	243	4	theorem	theorem	ADJ
ejpam-5887	243	5	6	6	NUM
ejpam-5887	243	6	,	,	PUNCT
ejpam-5887	243	7	d2(k1,n	d2(k1,n	NOUN
ejpam-5887	243	8	)	)	PUNCT
ejpam-5887	243	9	;	;	PUNCT
ejpam-5887	243	10	n	n	PRON
ejpam-5887	243	11	≥	≥	NOUN
ejpam-5887	243	12	5	5	NUM
ejpam-5887	243	13	is	be	AUX
ejpam-5887	243	14	not	not	PART
ejpam-5887	243	15	an	an	DET
ejpam-5887	243	16	edge	edge	NOUN
ejpam-5887	243	17	5	5	NUM
ejpam-5887	243	18	-	-	PUNCT
ejpam-5887	243	19	product	product	NOUN
ejpam-5887	243	20	cordial	cordial	ADJ
ejpam-5887	243	21	graph	graph	NOUN
ejpam-5887	243	22	.	.	PUNCT
ejpam-5887	244	1	3.2	3.2	NUM
ejpam-5887	244	2	.	.	PUNCT
ejpam-5887	244	3	splitting	splitting	NOUN
ejpam-5887	244	4	graph	graph	NOUN
ejpam-5887	244	5	of	of	ADP
ejpam-5887	244	6	star	star	NOUN
ejpam-5887	244	7	in	in	ADP
ejpam-5887	244	8	this	this	DET
ejpam-5887	244	9	subsection	subsection	NOUN
ejpam-5887	245	1	,	,	PUNCT
ejpam-5887	245	2	we	we	PRON
ejpam-5887	245	3	show	show	VERB
ejpam-5887	245	4	that	that	SCONJ
ejpam-5887	245	5	the	the	DET
ejpam-5887	245	6	splitting	splitting	NOUN
ejpam-5887	245	7	graph	graph	NOUN
ejpam-5887	245	8	of	of	ADP
ejpam-5887	245	9	a	a	DET
ejpam-5887	245	10	star	star	NOUN
ejpam-5887	245	11	graph	graph	NOUN
ejpam-5887	245	12	s′(k1,n	s′(k1,n	PROPN
ejpam-5887	245	13	)	)	PUNCT
ejpam-5887	245	14	does	do	AUX
ejpam-5887	245	15	not	not	PART
ejpam-5887	245	16	admit	admit	VERB
ejpam-5887	245	17	the	the	DET
ejpam-5887	245	18	edge	edge	NOUN
ejpam-5887	245	19	k	k	NOUN
ejpam-5887	245	20	-	-	PUNCT
ejpam-5887	245	21	product	product	NOUN
ejpam-5887	245	22	cordial	cordial	ADJ
ejpam-5887	245	23	labeling	labeling	NOUN
ejpam-5887	245	24	for	for	ADP
ejpam-5887	245	25	k	k	PROPN
ejpam-5887	245	26	≤	≤	PROPN
ejpam-5887	245	27	n.	n.	NOUN
ejpam-5887	245	28	also	also	ADV
ejpam-5887	245	29	,	,	PUNCT
ejpam-5887	245	30	we	we	PRON
ejpam-5887	245	31	study	study	VERB
ejpam-5887	245	32	the	the	DET
ejpam-5887	245	33	edge	edge	NOUN
ejpam-5887	245	34	k	k	NOUN
ejpam-5887	245	35	-	-	PUNCT
ejpam-5887	245	36	product	product	NOUN
ejpam-5887	245	37	cordial	cordial	ADJ
ejpam-5887	245	38	behavior	behavior	NOUN
ejpam-5887	245	39	of	of	ADP
ejpam-5887	245	40	s′(k1,n	s′(k1,n	ADJ
ejpam-5887	245	41	)	)	PUNCT
ejpam-5887	245	42	for	for	ADP
ejpam-5887	245	43	k	k	PROPN
ejpam-5887	245	44	=	=	SYM
ejpam-5887	245	45	3	3	NUM
ejpam-5887	245	46	,	,	PUNCT
ejpam-5887	245	47	4	4	NUM
ejpam-5887	245	48	,	,	PUNCT
ejpam-5887	245	49	5	5	NUM
ejpam-5887	245	50	.	.	X
ejpam-5887	245	51	theorem	theorem	VERB
ejpam-5887	245	52	10	10	NUM
ejpam-5887	245	53	.	.	PUNCT
ejpam-5887	246	1	the	the	DET
ejpam-5887	246	2	graph	graph	NOUN
ejpam-5887	246	3	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	246	4	)	)	PUNCT
ejpam-5887	246	5	dos	do	NOUN
ejpam-5887	246	6	not	not	PART
ejpam-5887	246	7	admit	admit	VERB
ejpam-5887	246	8	an	an	DET
ejpam-5887	246	9	edge	edge	NOUN
ejpam-5887	246	10	k	k	NOUN
ejpam-5887	246	11	-	-	PUNCT
ejpam-5887	246	12	product	product	NOUN
ejpam-5887	246	13	cordial	cordial	ADJ
ejpam-5887	246	14	labeling	labeling	NOUN
ejpam-5887	246	15	for	for	ADP
ejpam-5887	246	16	k	k	PROPN
ejpam-5887	246	17	≤	≤	PROPN
ejpam-5887	246	18	n.	n.	NOUN
ejpam-5887	246	19	proof	proof	NOUN
ejpam-5887	246	20	.	.	PUNCT
ejpam-5887	247	1	let	let	VERB
ejpam-5887	247	2	k	k	PROPN
ejpam-5887	247	3	≤	≤	PROPN
ejpam-5887	247	4	n.	n.	NOUN
ejpam-5887	247	5	we	we	PRON
ejpam-5887	247	6	consider	consider	VERB
ejpam-5887	247	7	the	the	DET
ejpam-5887	247	8	following	follow	VERB
ejpam-5887	247	9	two	two	NUM
ejpam-5887	247	10	cases	case	NOUN
ejpam-5887	247	11	.	.	PUNCT
ejpam-5887	248	1	case	case	NOUN
ejpam-5887	248	2	(	(	PUNCT
ejpam-5887	248	3	i	i	NOUN
ejpam-5887	248	4	):	):	PUNCT
ejpam-5887	248	5	for	for	ADP
ejpam-5887	248	6	n	n	NOUN
ejpam-5887	248	7	=	=	SYM
ejpam-5887	248	8	tk	tk	PROPN
ejpam-5887	249	1	+	+	CCONJ
ejpam-5887	249	2	k	k	PROPN
ejpam-5887	250	1	−	−	PROPN
ejpam-5887	250	2	1	1	NUM
ejpam-5887	250	3	,	,	PUNCT
ejpam-5887	250	4	we	we	PRON
ejpam-5887	250	5	have	have	AUX
ejpam-5887	250	6	|v	|v	VERB
ejpam-5887	250	7	|	|	ADV
ejpam-5887	251	1	=	=	SYM
ejpam-5887	251	2	2tk	2tk	PROPN
ejpam-5887	252	1	+	+	CCONJ
ejpam-5887	252	2	2k	2k	NUM
ejpam-5887	252	3	and	and	CCONJ
ejpam-5887	252	4	|e|	|e|	NOUN
ejpam-5887	252	5	=	=	NOUN
ejpam-5887	252	6	3tk	3tk	NOUN
ejpam-5887	253	1	+	+	PUNCT
ejpam-5887	253	2	3(k	3(k	NUM
ejpam-5887	253	3	−	−	NUM
ejpam-5887	253	4	1	1	NUM
ejpam-5887	253	5	)	)	PUNCT
ejpam-5887	253	6	.	.	PUNCT
ejpam-5887	254	1	clearly	clearly	ADV
ejpam-5887	254	2	,	,	PUNCT
ejpam-5887	254	3	|v	|v	PROPN
ejpam-5887	254	4	|	|	ADV
ejpam-5887	254	5	≡	≡	PROPN
ejpam-5887	254	6	0	0	PUNCT
ejpam-5887	255	1	(	(	PUNCT
ejpam-5887	255	2	mod	mod	PROPN
ejpam-5887	255	3	k	k	PROPN
ejpam-5887	255	4	)	)	PUNCT
ejpam-5887	256	1	and	and	CCONJ
ejpam-5887	256	2	k	k	X
ejpam-5887	256	3	<	<	X
ejpam-5887	256	4	|v	|v	PROPN
ejpam-5887	257	1	|	|	ADV
ejpam-5887	257	2	<	<	X
ejpam-5887	257	3	|e|	|e|	PROPN
ejpam-5887	257	4	.	.	PUNCT
ejpam-5887	257	5	by	by	ADP
ejpam-5887	257	6	theorem	theorem	NOUN
ejpam-5887	257	7	5	5	NUM
ejpam-5887	257	8	,	,	PUNCT
ejpam-5887	257	9	s′(kn	s′(kn	NOUN
ejpam-5887	257	10	)	)	PUNCT
ejpam-5887	257	11	is	be	AUX
ejpam-5887	257	12	not	not	PART
ejpam-5887	257	13	an	an	DET
ejpam-5887	257	14	edge	edge	NOUN
ejpam-5887	257	15	k	k	NOUN
ejpam-5887	257	16	-	-	PUNCT
ejpam-5887	257	17	product	product	NOUN
ejpam-5887	257	18	cordial	cordial	ADJ
ejpam-5887	257	19	graph	graph	NOUN
ejpam-5887	257	20	.	.	PUNCT
ejpam-5887	258	1	case	case	NOUN
ejpam-5887	258	2	(	(	PUNCT
ejpam-5887	258	3	ii	ii	NUM
ejpam-5887	258	4	):	):	PUNCT
ejpam-5887	258	5	for	for	ADP
ejpam-5887	258	6	n	n	NOUN
ejpam-5887	258	7	=	=	SYM
ejpam-5887	258	8	tk	tk	PROPN
ejpam-5887	259	1	+	+	CCONJ
ejpam-5887	259	2	r	r	NOUN
ejpam-5887	259	3	;	;	PUNCT
ejpam-5887	259	4	t	t	PROPN
ejpam-5887	259	5	≥	≥	NUM
ejpam-5887	259	6	1	1	NUM
ejpam-5887	259	7	and	and	CCONJ
ejpam-5887	259	8	0	0	NUM
ejpam-5887	259	9	≤	≤	NUM
ejpam-5887	259	10	r	r	NOUN
ejpam-5887	259	11	≤	≤	NUM
ejpam-5887	259	12	k	k	NOUN
ejpam-5887	260	1	−	−	PROPN
ejpam-5887	260	2	2	2	NUM
ejpam-5887	260	3	,	,	PUNCT
ejpam-5887	260	4	we	we	PRON
ejpam-5887	260	5	have	have	AUX
ejpam-5887	260	6	|v	|v	VERB
ejpam-5887	260	7	|	|	ADV
ejpam-5887	260	8	=	=	SYM
ejpam-5887	260	9	2tk	2tk	PROPN
ejpam-5887	261	1	+	+	CCONJ
ejpam-5887	261	2	2(r	2(r	NUM
ejpam-5887	261	3	+	+	NUM
ejpam-5887	261	4	1	1	NUM
ejpam-5887	261	5	)	)	PUNCT
ejpam-5887	261	6	and	and	CCONJ
ejpam-5887	261	7	|e|	|e|	PRON
ejpam-5887	261	8	=	=	NOUN
ejpam-5887	261	9	3tk	3tk	NOUN
ejpam-5887	262	1	+	+	NOUN
ejpam-5887	262	2	3r	3r	NUM
ejpam-5887	262	3	.	.	PUNCT
ejpam-5887	263	1	if	if	SCONJ
ejpam-5887	263	2	f	f	PROPN
ejpam-5887	263	3	is	be	AUX
ejpam-5887	263	4	an	an	DET
ejpam-5887	263	5	edge	edge	NOUN
ejpam-5887	263	6	k	k	NOUN
ejpam-5887	263	7	-	-	PUNCT
ejpam-5887	263	8	product	product	NOUN
ejpam-5887	263	9	cordial	cordial	ADJ
ejpam-5887	263	10	labeling	labeling	NOUN
ejpam-5887	263	11	of	of	ADP
ejpam-5887	263	12	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	263	13	)	)	PUNCT
ejpam-5887	263	14	,	,	PUNCT
ejpam-5887	263	15	then	then	ADV
ejpam-5887	263	16	ef	ef	PROPN
ejpam-5887	263	17	(	(	PUNCT
ejpam-5887	263	18	i	i	NOUN
ejpam-5887	263	19	)	)	PUNCT
ejpam-5887	263	20	=	=	PUNCT
ejpam-5887	263	21			NUM
ejpam-5887	263	22	3	3	NUM
ejpam-5887	263	23	t	t	NOUN
ejpam-5887	263	24	;	;	PUNCT
ejpam-5887	263	25	r	r	NOUN
ejpam-5887	263	26	=	=	SYM
ejpam-5887	263	27	0	0	NUM
ejpam-5887	263	28	3t+	3t+	NUM
ejpam-5887	263	29	1	1	NUM
ejpam-5887	263	30	;	;	PUNCT
ejpam-5887	263	31	r	r	NOUN
ejpam-5887	263	32	=	=	SYM
ejpam-5887	263	33	1	1	NUM
ejpam-5887	263	34	,	,	PUNCT
ejpam-5887	263	35	k	k	PROPN
ejpam-5887	263	36	=	=	SYM
ejpam-5887	263	37	3	3	NUM
ejpam-5887	263	38	3	3	NUM
ejpam-5887	263	39	t	t	NOUN
ejpam-5887	263	40	or	or	CCONJ
ejpam-5887	263	41	3t+	3t+	NUM
ejpam-5887	263	42	1	1	NUM
ejpam-5887	263	43	;	;	PUNCT
ejpam-5887	263	44	r	r	NOUN
ejpam-5887	263	45	=	=	SYM
ejpam-5887	263	46	1	1	NUM
ejpam-5887	263	47	,	,	PUNCT
ejpam-5887	263	48	k	k	X
ejpam-5887	263	49	≥	≥	NUM
ejpam-5887	263	50	4	4	NUM
ejpam-5887	263	51	3t+	3t+	NUM
ejpam-5887	263	52	⌊3rk	⌊3rk	PUNCT
ejpam-5887	263	53	⌋	⌋	NOUN
ejpam-5887	263	54	or	or	CCONJ
ejpam-5887	263	55	3t+	3t+	NUM
ejpam-5887	263	56	⌊3rk	⌊3rk	PUNCT
ejpam-5887	263	57	⌋+	⌋+	PUNCT
ejpam-5887	263	58	1	1	NUM
ejpam-5887	263	59	;	;	PUNCT
ejpam-5887	263	60	2	2	NUM
ejpam-5887	263	61	≤	≤	NOUN
ejpam-5887	263	62	r	r	NOUN
ejpam-5887	263	63	≤	≤	NUM
ejpam-5887	263	64	k	k	NOUN
ejpam-5887	264	1	−	−	PROPN
ejpam-5887	264	2	2	2	NUM
ejpam-5887	264	3	,	,	PUNCT
ejpam-5887	264	4	;	;	PUNCT
ejpam-5887	264	5	i	i	PRON
ejpam-5887	264	6	∈	∈	PROPN
ejpam-5887	264	7	{	{	PUNCT
ejpam-5887	264	8	0	0	NUM
ejpam-5887	264	9	,	,	PUNCT
ejpam-5887	264	10	1	1	NUM
ejpam-5887	264	11	,	,	PUNCT
ejpam-5887	264	12	...	...	PUNCT
ejpam-5887	264	13	,	,	PUNCT
ejpam-5887	264	14	k	k	PROPN
ejpam-5887	265	1	−	−	PROPN
ejpam-5887	265	2	1	1	NUM
ejpam-5887	265	3	}	}	PUNCT
ejpam-5887	265	4	,	,	PUNCT
ejpam-5887	265	5	vf∗(i	vf∗(i	ADJ
ejpam-5887	265	6	)	)	PUNCT
ejpam-5887	265	7	=	=	PUNCT
ejpam-5887	265	8			NOUN
ejpam-5887	265	9	2t+	2t+	NUM
ejpam-5887	265	10	1	1	NUM
ejpam-5887	265	11	;	;	PUNCT
ejpam-5887	265	12	r	r	NOUN
ejpam-5887	265	13	=	=	SYM
ejpam-5887	265	14	0	0	NUM
ejpam-5887	265	15	,	,	PUNCT
ejpam-5887	265	16	k	k	PROPN
ejpam-5887	265	17	=	=	SYM
ejpam-5887	265	18	2	2	NUM
ejpam-5887	265	19	2t+	2t+	NUM
ejpam-5887	265	20	1	1	NUM
ejpam-5887	265	21	or	or	CCONJ
ejpam-5887	265	22	2t+	2t+	NUM
ejpam-5887	265	23	2	2	NUM
ejpam-5887	265	24	;	;	PUNCT
ejpam-5887	265	25	r	r	NOUN
ejpam-5887	265	26	=	=	SYM
ejpam-5887	265	27	1	1	NUM
ejpam-5887	265	28	,	,	PUNCT
ejpam-5887	265	29	k	k	PROPN
ejpam-5887	265	30	=	=	SYM
ejpam-5887	265	31	3	3	NUM
ejpam-5887	265	32	2t+	2t+	NUM
ejpam-5887	265	33	1	1	NUM
ejpam-5887	265	34	;	;	PUNCT
ejpam-5887	265	35	r	r	NOUN
ejpam-5887	265	36	=	=	SYM
ejpam-5887	265	37	1	1	NUM
ejpam-5887	265	38	,	,	PUNCT
ejpam-5887	265	39	k	k	NOUN
ejpam-5887	265	40	=	=	NOUN
ejpam-5887	265	41	4	4	NUM
ejpam-5887	265	42	2	2	NUM
ejpam-5887	265	43	t	t	NOUN
ejpam-5887	265	44	or	or	CCONJ
ejpam-5887	265	45	2t+	2t+	NUM
ejpam-5887	265	46	1	1	NUM
ejpam-5887	265	47	;	;	PUNCT
ejpam-5887	265	48	r	r	NOUN
ejpam-5887	265	49	=	=	SYM
ejpam-5887	265	50	0	0	NUM
ejpam-5887	265	51	,	,	PUNCT
ejpam-5887	265	52	k	k	PROPN
ejpam-5887	265	53	≥	≥	NUM
ejpam-5887	265	54	3	3	NUM
ejpam-5887	265	55	;	;	PUNCT
ejpam-5887	265	56	r	r	NOUN
ejpam-5887	265	57	=	=	SYM
ejpam-5887	265	58	1	1	NUM
ejpam-5887	265	59	,	,	PUNCT
ejpam-5887	265	60	k	k	X
ejpam-5887	265	61	≥	≥	NUM
ejpam-5887	265	62	5	5	NUM
ejpam-5887	265	63	2t+	2t+	NUM
ejpam-5887	265	64	⌊2r+2	⌊2r+2	X
ejpam-5887	265	65	k	k	X
ejpam-5887	265	66	⌋	⌋	NOUN
ejpam-5887	265	67	or	or	CCONJ
ejpam-5887	265	68	2t+	2t+	NUM
ejpam-5887	265	69	⌊2r+2	⌊2r+2	X
ejpam-5887	266	1	k	k	X
ejpam-5887	266	2	⌋+	⌋+	ADJ
ejpam-5887	266	3	1	1	NUM
ejpam-5887	266	4	;	;	PUNCT
ejpam-5887	266	5	2	2	NUM
ejpam-5887	266	6	≤	≤	NOUN
ejpam-5887	266	7	r	r	NOUN
ejpam-5887	266	8	≤	≤	PUNCT
ejpam-5887	266	9	k	k	NOUN
ejpam-5887	266	10	−	−	PROPN
ejpam-5887	266	11	2	2	NUM
ejpam-5887	266	12	.	.	PUNCT
ejpam-5887	266	13	;	;	PUNCT
ejpam-5887	266	14	i	i	PRON
ejpam-5887	266	15	∈	∈	PROPN
ejpam-5887	266	16	{	{	PUNCT
ejpam-5887	266	17	0	0	NUM
ejpam-5887	266	18	,	,	PUNCT
ejpam-5887	266	19	1	1	NUM
ejpam-5887	266	20	,	,	PUNCT
ejpam-5887	266	21	..	..	PUNCT
ejpam-5887	266	22	,	,	PUNCT
ejpam-5887	266	23	k	k	PROPN
ejpam-5887	267	1	−	−	PROPN
ejpam-5887	267	2	1	1	NUM
ejpam-5887	267	3	}	}	PUNCT
ejpam-5887	267	4	.	.	PUNCT
ejpam-5887	268	1	for	for	ADP
ejpam-5887	268	2	the	the	DET
ejpam-5887	268	3	case	case	NOUN
ejpam-5887	268	4	where	where	SCONJ
ejpam-5887	268	5	0	0	NUM
ejpam-5887	268	6	≤	≤	NUM
ejpam-5887	268	7	r	r	NOUN
ejpam-5887	268	8	≤	≤	NUM
ejpam-5887	268	9	1	1	NUM
ejpam-5887	268	10	,	,	PUNCT
ejpam-5887	268	11	ef	ef	X
ejpam-5887	268	12	(	(	PUNCT
ejpam-5887	268	13	0	0	NUM
ejpam-5887	268	14	)	)	PUNCT
ejpam-5887	268	15	=	=	SYM
ejpam-5887	268	16	3	3	NUM
ejpam-5887	268	17	t	t	NOUN
ejpam-5887	268	18	implies	imply	VERB
ejpam-5887	268	19	vf∗(0	vf∗(0	NOUN
ejpam-5887	268	20	)	)	PUNCT
ejpam-5887	268	21	≥	≥	NOUN
ejpam-5887	268	22	3t+	3t+	NUM
ejpam-5887	268	23	1	1	NUM
ejpam-5887	268	24	>	>	SYM
ejpam-5887	268	25	2t+	2t+	NUM
ejpam-5887	268	26	1	1	NUM
ejpam-5887	268	27	for	for	ADP
ejpam-5887	268	28	t	t	PROPN
ejpam-5887	268	29	≥	≥	NUM
ejpam-5887	268	30	1	1	NUM
ejpam-5887	268	31	.	.	PUNCT
ejpam-5887	269	1	also	also	ADV
ejpam-5887	269	2	,	,	PUNCT
ejpam-5887	269	3	ef	ef	PROPN
ejpam-5887	269	4	(	(	PUNCT
ejpam-5887	269	5	0	0	NUM
ejpam-5887	269	6	)	)	PUNCT
ejpam-5887	269	7	=	=	NOUN
ejpam-5887	269	8	3t+	3t+	NUM
ejpam-5887	269	9	1	1	NUM
ejpam-5887	269	10	implies	imply	VERB
ejpam-5887	269	11	vf∗(0	vf∗(0	NOUN
ejpam-5887	269	12	)	)	PUNCT
ejpam-5887	269	13	≥	≥	NOUN
ejpam-5887	269	14	3t+	3t+	NUM
ejpam-5887	269	15	2	2	NUM
ejpam-5887	269	16	>	>	SYM
ejpam-5887	269	17	2t+	2t+	NUM
ejpam-5887	269	18	2	2	NUM
ejpam-5887	269	19	for	for	ADP
ejpam-5887	269	20	t	t	PROPN
ejpam-5887	269	21	≥	≥	NUM
ejpam-5887	269	22	1	1	NUM
ejpam-5887	269	23	.	.	PUNCT
ejpam-5887	270	1	therefore	therefore	ADV
ejpam-5887	270	2	,	,	PUNCT
ejpam-5887	270	3	|vf∗(0)−	|vf∗(0)−	PROPN
ejpam-5887	270	4	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	270	5	>	>	X
ejpam-5887	270	6	1	1	NUM
ejpam-5887	270	7	for	for	ADP
ejpam-5887	270	8	some	some	PRON
ejpam-5887	270	9	j	j	NOUN
ejpam-5887	270	10	=	=	SYM
ejpam-5887	270	11	1	1	NUM
ejpam-5887	270	12	,	,	PUNCT
ejpam-5887	270	13	2	2	NUM
ejpam-5887	270	14	,	,	PUNCT
ejpam-5887	270	15	...	...	PUNCT
ejpam-5887	270	16	,	,	PUNCT
ejpam-5887	270	17	k−1	k−1	PROPN
ejpam-5887	270	18	,	,	PUNCT
ejpam-5887	270	19	which	which	PRON
ejpam-5887	270	20	is	be	AUX
ejpam-5887	270	21	a	a	DET
ejpam-5887	270	22	contradiction	contradiction	NOUN
ejpam-5887	270	23	.	.	PUNCT
ejpam-5887	271	1	in	in	ADP
ejpam-5887	271	2	the	the	DET
ejpam-5887	271	3	other	other	ADJ
ejpam-5887	271	4	cases	case	NOUN
ejpam-5887	271	5	,	,	PUNCT
ejpam-5887	271	6	if	if	SCONJ
ejpam-5887	271	7	ef	ef	X
ejpam-5887	271	8	(	(	PUNCT
ejpam-5887	271	9	0	0	NUM
ejpam-5887	271	10	)	)	PUNCT
ejpam-5887	271	11	=	=	SYM
ejpam-5887	271	12	3t+⌊3rk	3t+⌊3rk	NUM
ejpam-5887	271	13	⌋	⌋	NOUN
ejpam-5887	271	14	,	,	PUNCT
ejpam-5887	271	15	then	then	ADV
ejpam-5887	271	16	vf∗(0	vf∗(0	VERB
ejpam-5887	271	17	)	)	PUNCT
ejpam-5887	271	18	≥	≥	NOUN
ejpam-5887	271	19	3	3	NUM
ejpam-5887	271	20	t	t	NOUN
ejpam-5887	271	21	+	+	CCONJ
ejpam-5887	271	22	⌊3rk	⌊3rk	X
ejpam-5887	271	23	⌋	⌋	NOUN
ejpam-5887	272	1	+	+	CCONJ
ejpam-5887	272	2	1	1	X
ejpam-5887	272	3	.	.	PUNCT
ejpam-5887	272	4	since	since	SCONJ
ejpam-5887	272	5	⌊3rk	⌊3rk	PROPN
ejpam-5887	272	6	⌋	⌋	PROPN
ejpam-5887	272	7	≥	≥	NUM
ejpam-5887	272	8	⌊2r+2	⌊2r+2	X
ejpam-5887	272	9	k	k	X
ejpam-5887	272	10	⌋	⌋	NOUN
ejpam-5887	272	11	for	for	ADP
ejpam-5887	272	12	2	2	NUM
ejpam-5887	272	13	≤	≤	NOUN
ejpam-5887	272	14	r	r	NOUN
ejpam-5887	272	15	≤	≤	NUM
ejpam-5887	272	16	k	k	NOUN
ejpam-5887	273	1	−	−	PROPN
ejpam-5887	273	2	2	2	NUM
ejpam-5887	273	3	,	,	PUNCT
ejpam-5887	273	4	we	we	PRON
ejpam-5887	273	5	have	have	VERB
ejpam-5887	273	6	vf∗(0	vf∗(0	NOUN
ejpam-5887	273	7	)	)	PUNCT
ejpam-5887	273	8	>	>	X
ejpam-5887	273	9	2t+	2t+	NUM
ejpam-5887	273	10	⌊2r+2	⌊2r+2	X
ejpam-5887	273	11	k	k	X
ejpam-5887	273	12	⌋+1	⌋+1	PROPN
ejpam-5887	273	13	for	for	ADP
ejpam-5887	273	14	t	t	PROPN
ejpam-5887	273	15	≥	≥	NUM
ejpam-5887	273	16	1	1	NUM
ejpam-5887	273	17	.	.	PUNCT
ejpam-5887	274	1	therefore	therefore	ADV
ejpam-5887	274	2	,	,	PUNCT
ejpam-5887	274	3	|vf∗(0)−	|vf∗(0)−	PROPN
ejpam-5887	274	4	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	274	5	>	>	X
ejpam-5887	274	6	1	1	NUM
ejpam-5887	274	7	for	for	ADP
ejpam-5887	274	8	some	some	PRON
ejpam-5887	274	9	j	j	NOUN
ejpam-5887	274	10	=	=	SYM
ejpam-5887	274	11	1	1	NUM
ejpam-5887	274	12	,	,	PUNCT
ejpam-5887	274	13	2	2	NUM
ejpam-5887	274	14	,	,	PUNCT
ejpam-5887	274	15	..	..	PUNCT
ejpam-5887	274	16	,	,	PUNCT
ejpam-5887	274	17	k−	k−	PROPN
ejpam-5887	274	18	1	1	NUM
ejpam-5887	274	19	,	,	PUNCT
ejpam-5887	274	20	which	which	PRON
ejpam-5887	274	21	is	be	AUX
ejpam-5887	274	22	a	a	DET
ejpam-5887	274	23	contradiction	contradiction	NOUN
ejpam-5887	274	24	.	.	PUNCT
ejpam-5887	275	1	hence	hence	ADV
ejpam-5887	275	2	,	,	PUNCT
ejpam-5887	275	3	s′(k1,n	s′(k1,n	X
ejpam-5887	275	4	)	)	PUNCT
ejpam-5887	275	5	is	be	AUX
ejpam-5887	275	6	not	not	PART
ejpam-5887	275	7	an	an	DET
ejpam-5887	275	8	edge	edge	NOUN
ejpam-5887	275	9	k	k	NOUN
ejpam-5887	275	10	-	-	PUNCT
ejpam-5887	275	11	product	product	NOUN
ejpam-5887	275	12	cordial	cordial	ADJ
ejpam-5887	275	13	graph	graph	NOUN
ejpam-5887	275	14	if	if	SCONJ
ejpam-5887	275	15	k	k	PROPN
ejpam-5887	275	16	≤	≤	PROPN
ejpam-5887	275	17	n.	n.	NOUN
ejpam-5887	275	18	theorem	theorem	VERB
ejpam-5887	275	19	11	11	NUM
ejpam-5887	275	20	.	.	PUNCT
ejpam-5887	276	1	the	the	DET
ejpam-5887	276	2	graph	graph	NOUN
ejpam-5887	276	3	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	276	4	)	)	PUNCT
ejpam-5887	276	5	admits	admit	VERB
ejpam-5887	276	6	an	an	DET
ejpam-5887	276	7	edge	edge	NOUN
ejpam-5887	276	8	3	3	NUM
ejpam-5887	276	9	-	-	PUNCT
ejpam-5887	276	10	product	product	NOUN
ejpam-5887	276	11	cordial	cordial	ADJ
ejpam-5887	276	12	labeling	labeling	NOUN
ejpam-5887	276	13	if	if	SCONJ
ejpam-5887	276	14	and	and	CCONJ
ejpam-5887	276	15	only	only	ADV
ejpam-5887	276	16	if	if	SCONJ
ejpam-5887	276	17	n	n	PROPN
ejpam-5887	276	18	=	=	SYM
ejpam-5887	276	19	1	1	X
ejpam-5887	276	20	.	.	PUNCT
ejpam-5887	277	1	proof	proof	NOUN
ejpam-5887	277	2	.	.	PUNCT
ejpam-5887	278	1	let	let	AUX
ejpam-5887	278	2	let	let	VERB
ejpam-5887	278	3	the	the	DET
ejpam-5887	278	4	vertex	vertex	NOUN
ejpam-5887	278	5	set	set	NOUN
ejpam-5887	278	6	and	and	CCONJ
ejpam-5887	278	7	edge	edge	NOUN
ejpam-5887	278	8	set	set	NOUN
ejpam-5887	278	9	of	of	ADP
ejpam-5887	278	10	s′(k1,n	s′(k1,n	ADJ
ejpam-5887	278	11	)	)	PUNCT
ejpam-5887	278	12	be	be	AUX
ejpam-5887	278	13	v	v	ADP
ejpam-5887	278	14	(	(	PUNCT
ejpam-5887	278	15	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	278	16	)	)	PUNCT
ejpam-5887	278	17	)	)	PUNCT
ejpam-5887	279	1	=	=	PRON
ejpam-5887	279	2	{	{	PUNCT
ejpam-5887	279	3	u	u	PROPN
ejpam-5887	279	4	,	,	PUNCT
ejpam-5887	279	5	ui	ui	PROPN
ejpam-5887	279	6	,	,	PUNCT
ejpam-5887	279	7	v	v	NOUN
ejpam-5887	279	8	,	,	PUNCT
ejpam-5887	279	9	vi	vi	NOUN
ejpam-5887	279	10	;	;	PUNCT
ejpam-5887	279	11	1	1	NUM
ejpam-5887	279	12	≤	≤	NUM
ejpam-5887	279	13	i	i	PRON
ejpam-5887	279	14	≤	≤	NOUN
ejpam-5887	279	15	n	n	CCONJ
ejpam-5887	279	16	}	}	PUNCT
ejpam-5887	279	17	and	and	CCONJ
ejpam-5887	279	18	e(s′(k1,n	e(s′(k1,n	NOUN
ejpam-5887	279	19	)	)	PUNCT
ejpam-5887	279	20	)	)	PUNCT
ejpam-5887	280	1	=	=	PRON
ejpam-5887	280	2	{	{	PUNCT
ejpam-5887	280	3	uui	uui	PROPN
ejpam-5887	280	4	,	,	PUNCT
ejpam-5887	280	5	vui	vui	PROPN
ejpam-5887	280	6	,	,	PUNCT
ejpam-5887	280	7	uvi	uvi	PROPN
ejpam-5887	280	8	;	;	PUNCT
ejpam-5887	280	9	1	1	NUM
ejpam-5887	280	10	≤	≤	NUM
ejpam-5887	280	11	i	i	PRON
ejpam-5887	280	12	≤	≤	NOUN
ejpam-5887	280	13	n	n	CCONJ
ejpam-5887	280	14	}	}	PUNCT
ejpam-5887	280	15	respectively	respectively	ADV
ejpam-5887	280	16	.	.	PUNCT
ejpam-5887	281	1	define	define	VERB
ejpam-5887	281	2	an	an	DET
ejpam-5887	281	3	edge	edge	NOUN
ejpam-5887	281	4	labeling	labeling	NOUN
ejpam-5887	281	5	f	f	NOUN
ejpam-5887	281	6	:	:	PUNCT
ejpam-5887	281	7	e(s′(k1,1	e(s′(k1,1	NUM
ejpam-5887	281	8	)	)	PUNCT
ejpam-5887	281	9	)	)	PUNCT
ejpam-5887	282	1	→	→	PUNCT
ejpam-5887	282	2	{	{	PUNCT
ejpam-5887	282	3	0	0	NUM
ejpam-5887	282	4	,	,	PUNCT
ejpam-5887	282	5	1	1	NUM
ejpam-5887	282	6	,	,	PUNCT
ejpam-5887	282	7	2	2	NUM
ejpam-5887	282	8	}	}	PUNCT
ejpam-5887	282	9	as	as	SCONJ
ejpam-5887	282	10	follows	follow	VERB
ejpam-5887	282	11	:	:	PUNCT
ejpam-5887	282	12	n.	n.	PROPN
ejpam-5887	282	13	m.	m.	NOUN
ejpam-5887	282	14	noureldeen	noureldeen	NOUN
ejpam-5887	282	15	et	et	PROPN
ejpam-5887	282	16	al	al	PROPN
ejpam-5887	282	17	.	.	PUNCT
ejpam-5887	282	18	/	/	SYM
ejpam-5887	282	19	eur	eur	PROPN
ejpam-5887	282	20	.	.	PUNCT
ejpam-5887	283	1	j.	j.	PROPN
ejpam-5887	283	2	pure	pure	PROPN
ejpam-5887	283	3	appl	appl	PROPN
ejpam-5887	283	4	.	.	PROPN
ejpam-5887	283	5	math	math	PROPN
ejpam-5887	283	6	,	,	PUNCT
ejpam-5887	283	7	18	18	NUM
ejpam-5887	283	8	(	(	PUNCT
ejpam-5887	283	9	2	2	NUM
ejpam-5887	283	10	)	)	PUNCT
ejpam-5887	283	11	(	(	PUNCT
ejpam-5887	283	12	2025	2025	NUM
ejpam-5887	283	13	)	)	PUNCT
ejpam-5887	283	14	,	,	PUNCT
ejpam-5887	283	15	5887	5887	NUM
ejpam-5887	283	16	9	9	NUM
ejpam-5887	283	17	of	of	ADP
ejpam-5887	283	18	21	21	NUM
ejpam-5887	283	19	f(uu1	f(uu1	NOUN
ejpam-5887	283	20	)	)	PUNCT
ejpam-5887	283	21	=	=	SYM
ejpam-5887	283	22	0	0	NUM
ejpam-5887	283	23	,	,	PUNCT
ejpam-5887	283	24	f(vu1	f(vu1	NOUN
ejpam-5887	283	25	)	)	PUNCT
ejpam-5887	283	26	=	=	SYM
ejpam-5887	283	27	1	1	NUM
ejpam-5887	283	28	,	,	PUNCT
ejpam-5887	283	29	f(uv1	f(uv1	NOUN
ejpam-5887	283	30	)	)	PUNCT
ejpam-5887	283	31	=	=	SYM
ejpam-5887	284	1	2	2	X
ejpam-5887	284	2	.	.	X
ejpam-5887	284	3	from	from	ADP
ejpam-5887	284	4	this	this	DET
ejpam-5887	284	5	labeling	labeling	NOUN
ejpam-5887	284	6	we	we	PRON
ejpam-5887	284	7	get	get	VERB
ejpam-5887	284	8	,	,	PUNCT
ejpam-5887	284	9	ef	ef	PROPN
ejpam-5887	284	10	(	(	PUNCT
ejpam-5887	284	11	0	0	NUM
ejpam-5887	284	12	)	)	PUNCT
ejpam-5887	284	13	=	=	SYM
ejpam-5887	284	14	ef	ef	X
ejpam-5887	284	15	(	(	PUNCT
ejpam-5887	284	16	1	1	NUM
ejpam-5887	284	17	)	)	PUNCT
ejpam-5887	284	18	=	=	SYM
ejpam-5887	284	19	ef	ef	X
ejpam-5887	284	20	(	(	PUNCT
ejpam-5887	284	21	2	2	NUM
ejpam-5887	284	22	)	)	PUNCT
ejpam-5887	284	23	=	=	SYM
ejpam-5887	284	24	1	1	NUM
ejpam-5887	284	25	and	and	CCONJ
ejpam-5887	284	26	vf∗(0)−1	vf∗(0)−1	NOUN
ejpam-5887	284	27	=	=	SYM
ejpam-5887	284	28	vf∗(1	vf∗(1	X
ejpam-5887	284	29	)	)	PUNCT
ejpam-5887	284	30	=	=	SYM
ejpam-5887	284	31	vf∗(2	vf∗(2	ADJ
ejpam-5887	284	32	)	)	PUNCT
ejpam-5887	284	33	=	=	SYM
ejpam-5887	285	1	1	1	X
ejpam-5887	285	2	.	.	PUNCT
ejpam-5887	285	3	hence	hence	ADV
ejpam-5887	285	4	,	,	PUNCT
ejpam-5887	285	5	s′(k1,1	s′(k1,1	CCONJ
ejpam-5887	285	6	)	)	PUNCT
ejpam-5887	285	7	is	be	AUX
ejpam-5887	285	8	an	an	DET
ejpam-5887	285	9	edge	edge	NOUN
ejpam-5887	285	10	3	3	NUM
ejpam-5887	285	11	-	-	PUNCT
ejpam-5887	285	12	product	product	NOUN
ejpam-5887	285	13	cordial	cordial	ADJ
ejpam-5887	285	14	graph	graph	NOUN
ejpam-5887	285	15	.	.	PUNCT
ejpam-5887	286	1	for	for	ADP
ejpam-5887	286	2	n	n	NOUN
ejpam-5887	286	3	=	=	SYM
ejpam-5887	286	4	2	2	NUM
ejpam-5887	286	5	,	,	PUNCT
ejpam-5887	286	6	|v	|v	ADV
ejpam-5887	286	7	|	|	NOUN
ejpam-5887	287	1	=	=	SYM
ejpam-5887	287	2	|e|	|e|	NOUN
ejpam-5887	287	3	=	=	SYM
ejpam-5887	287	4	6	6	NUM
ejpam-5887	287	5	.	.	PUNCT
ejpam-5887	287	6	by	by	ADP
ejpam-5887	287	7	theorem	theorem	NOUN
ejpam-5887	287	8	5	5	NUM
ejpam-5887	287	9	,	,	PUNCT
ejpam-5887	287	10	s′(k1,2	s′(k1,2	ADJ
ejpam-5887	287	11	)	)	PUNCT
ejpam-5887	287	12	is	be	AUX
ejpam-5887	287	13	not	not	PART
ejpam-5887	287	14	an	an	DET
ejpam-5887	287	15	edge	edge	NOUN
ejpam-5887	287	16	3	3	NUM
ejpam-5887	287	17	-	-	PUNCT
ejpam-5887	287	18	product	product	NOUN
ejpam-5887	287	19	cordial	cordial	ADJ
ejpam-5887	287	20	graph	graph	NOUN
ejpam-5887	287	21	.	.	PUNCT
ejpam-5887	288	1	also	also	ADV
ejpam-5887	288	2	,	,	PUNCT
ejpam-5887	288	3	by	by	ADP
ejpam-5887	288	4	theorem	theorem	NOUN
ejpam-5887	288	5	10	10	NUM
ejpam-5887	288	6	,	,	PUNCT
ejpam-5887	288	7	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	288	8	)	)	PUNCT
ejpam-5887	288	9	;	;	PUNCT
ejpam-5887	288	10	n	n	PRON
ejpam-5887	288	11	≥	≥	NOUN
ejpam-5887	288	12	3	3	NUM
ejpam-5887	288	13	is	be	AUX
ejpam-5887	288	14	not	not	PART
ejpam-5887	288	15	an	an	DET
ejpam-5887	288	16	edge	edge	NOUN
ejpam-5887	288	17	3	3	NUM
ejpam-5887	288	18	-	-	PUNCT
ejpam-5887	288	19	product	product	NOUN
ejpam-5887	288	20	cordial	cordial	ADJ
ejpam-5887	288	21	graph	graph	NOUN
ejpam-5887	288	22	.	.	PUNCT
ejpam-5887	289	1	theorem	theorem	NOUN
ejpam-5887	289	2	12	12	NUM
ejpam-5887	289	3	.	.	PUNCT
ejpam-5887	290	1	the	the	DET
ejpam-5887	290	2	graph	graph	NOUN
ejpam-5887	290	3	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	290	4	)	)	PUNCT
ejpam-5887	290	5	admits	admit	VERB
ejpam-5887	290	6	an	an	DET
ejpam-5887	290	7	edge	edge	NOUN
ejpam-5887	290	8	4	4	NUM
ejpam-5887	290	9	-	-	PUNCT
ejpam-5887	290	10	product	product	NOUN
ejpam-5887	290	11	cordial	cordial	ADJ
ejpam-5887	290	12	labeling	labeling	NOUN
ejpam-5887	290	13	if	if	SCONJ
ejpam-5887	290	14	and	and	CCONJ
ejpam-5887	290	15	only	only	ADV
ejpam-5887	290	16	if	if	SCONJ
ejpam-5887	290	17	n	n	NOUN
ejpam-5887	290	18	=	=	SYM
ejpam-5887	290	19	2	2	X
ejpam-5887	290	20	.	.	PUNCT
ejpam-5887	291	1	proof	proof	NOUN
ejpam-5887	291	2	.	.	PUNCT
ejpam-5887	292	1	let	let	VERB
ejpam-5887	292	2	the	the	DET
ejpam-5887	292	3	vertex	vertex	NOUN
ejpam-5887	292	4	set	set	NOUN
ejpam-5887	292	5	and	and	CCONJ
ejpam-5887	292	6	edge	edge	NOUN
ejpam-5887	292	7	set	set	NOUN
ejpam-5887	292	8	of	of	ADP
ejpam-5887	292	9	s′(k1,n	s′(k1,n	ADJ
ejpam-5887	292	10	)	)	PUNCT
ejpam-5887	292	11	be	be	AUX
ejpam-5887	292	12	v	v	ADP
ejpam-5887	292	13	(	(	PUNCT
ejpam-5887	292	14	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	292	15	)	)	PUNCT
ejpam-5887	292	16	)	)	PUNCT
ejpam-5887	293	1	=	=	PRON
ejpam-5887	293	2	{	{	PUNCT
ejpam-5887	293	3	u	u	PROPN
ejpam-5887	293	4	,	,	PUNCT
ejpam-5887	293	5	ui	ui	PROPN
ejpam-5887	293	6	,	,	PUNCT
ejpam-5887	293	7	v	v	NOUN
ejpam-5887	293	8	,	,	PUNCT
ejpam-5887	293	9	vi	vi	NOUN
ejpam-5887	293	10	;	;	PUNCT
ejpam-5887	293	11	1	1	NUM
ejpam-5887	293	12	≤	≤	NUM
ejpam-5887	293	13	i	i	PRON
ejpam-5887	293	14	≤	≤	NOUN
ejpam-5887	293	15	n	n	CCONJ
ejpam-5887	293	16	}	}	PUNCT
ejpam-5887	293	17	and	and	CCONJ
ejpam-5887	293	18	e(s′(k1,n	e(s′(k1,n	NOUN
ejpam-5887	293	19	)	)	PUNCT
ejpam-5887	293	20	)	)	PUNCT
ejpam-5887	294	1	=	=	PRON
ejpam-5887	294	2	{	{	PUNCT
ejpam-5887	294	3	uui	uui	PROPN
ejpam-5887	294	4	,	,	PUNCT
ejpam-5887	294	5	vui	vui	PROPN
ejpam-5887	294	6	,	,	PUNCT
ejpam-5887	294	7	uvi	uvi	PROPN
ejpam-5887	294	8	;	;	PUNCT
ejpam-5887	294	9	1	1	NUM
ejpam-5887	294	10	≤	≤	NUM
ejpam-5887	294	11	i	i	PRON
ejpam-5887	294	12	≤	≤	NOUN
ejpam-5887	294	13	n	n	CCONJ
ejpam-5887	294	14	}	}	PUNCT
ejpam-5887	294	15	respectively	respectively	ADV
ejpam-5887	294	16	.	.	PUNCT
ejpam-5887	295	1	define	define	VERB
ejpam-5887	295	2	an	an	DET
ejpam-5887	295	3	edge	edge	NOUN
ejpam-5887	295	4	labeling	labeling	NOUN
ejpam-5887	295	5	f	f	NOUN
ejpam-5887	295	6	:	:	PUNCT
ejpam-5887	295	7	e(s′(k1,2	e(s′(k1,2	NUM
ejpam-5887	295	8	)	)	PUNCT
ejpam-5887	295	9	)	)	PUNCT
ejpam-5887	296	1	→	→	PUNCT
ejpam-5887	296	2	{	{	PUNCT
ejpam-5887	296	3	0	0	NUM
ejpam-5887	296	4	,	,	PUNCT
ejpam-5887	296	5	1	1	NUM
ejpam-5887	296	6	,	,	PUNCT
ejpam-5887	296	7	2	2	NUM
ejpam-5887	296	8	,	,	PUNCT
ejpam-5887	296	9	3	3	NUM
ejpam-5887	296	10	}	}	PUNCT
ejpam-5887	296	11	as	as	SCONJ
ejpam-5887	296	12	follows	follow	VERB
ejpam-5887	296	13	:	:	PUNCT
ejpam-5887	296	14	f(uu1	f(uu1	NOUN
ejpam-5887	296	15	)	)	PUNCT
ejpam-5887	296	16	=	=	SYM
ejpam-5887	296	17	0	0	NUM
ejpam-5887	296	18	,	,	PUNCT
ejpam-5887	296	19	f(uu2	f(uu2	ADJ
ejpam-5887	296	20	)	)	PUNCT
ejpam-5887	296	21	=	=	SYM
ejpam-5887	296	22	1	1	NUM
ejpam-5887	296	23	,	,	PUNCT
ejpam-5887	296	24	f(vu1	f(vu1	NOUN
ejpam-5887	296	25	)	)	PUNCT
ejpam-5887	296	26	=	=	SYM
ejpam-5887	296	27	3	3	NUM
ejpam-5887	296	28	,	,	PUNCT
ejpam-5887	296	29	f(vu2	f(vu2	NOUN
ejpam-5887	296	30	)	)	PUNCT
ejpam-5887	296	31	=	=	SYM
ejpam-5887	296	32	1	1	NUM
ejpam-5887	296	33	,	,	PUNCT
ejpam-5887	296	34	f(uv1	f(uv1	NOUN
ejpam-5887	296	35	)	)	PUNCT
ejpam-5887	296	36	=	=	SYM
ejpam-5887	296	37	2	2	NUM
ejpam-5887	296	38	,	,	PUNCT
ejpam-5887	296	39	f(uv2	f(uv2	ADJ
ejpam-5887	296	40	)	)	PUNCT
ejpam-5887	296	41	=	=	SYM
ejpam-5887	297	1	2	2	X
ejpam-5887	297	2	.	.	X
ejpam-5887	297	3	from	from	ADP
ejpam-5887	297	4	this	this	DET
ejpam-5887	297	5	labeling	labeling	NOUN
ejpam-5887	297	6	we	we	PRON
ejpam-5887	297	7	get	get	VERB
ejpam-5887	297	8	,	,	PUNCT
ejpam-5887	297	9	ef	ef	PROPN
ejpam-5887	297	10	(	(	PUNCT
ejpam-5887	297	11	0	0	NUM
ejpam-5887	297	12	)	)	PUNCT
ejpam-5887	297	13	=	=	SYM
ejpam-5887	297	14	ef	ef	X
ejpam-5887	297	15	(	(	PUNCT
ejpam-5887	297	16	1	1	NUM
ejpam-5887	297	17	)	)	PUNCT
ejpam-5887	297	18	−	−	PROPN
ejpam-5887	297	19	1	1	NUM
ejpam-5887	297	20	=	=	SYM
ejpam-5887	297	21	ef	ef	X
ejpam-5887	297	22	(	(	PUNCT
ejpam-5887	297	23	2	2	NUM
ejpam-5887	297	24	)	)	PUNCT
ejpam-5887	297	25	−	−	NOUN
ejpam-5887	297	26	1	1	NUM
ejpam-5887	297	27	=	=	SYM
ejpam-5887	297	28	ef	ef	X
ejpam-5887	297	29	(	(	PUNCT
ejpam-5887	297	30	3	3	NUM
ejpam-5887	297	31	)	)	PUNCT
ejpam-5887	297	32	=	=	SYM
ejpam-5887	297	33	1	1	NUM
ejpam-5887	297	34	and	and	CCONJ
ejpam-5887	297	35	vf∗(0	vf∗(0	NOUN
ejpam-5887	297	36	)	)	PUNCT
ejpam-5887	297	37	−	−	PROPN
ejpam-5887	297	38	1	1	NUM
ejpam-5887	297	39	=	=	NOUN
ejpam-5887	297	40	vf∗(1	vf∗(1	X
ejpam-5887	297	41	)	)	PUNCT
ejpam-5887	297	42	=	=	SYM
ejpam-5887	298	1	vf∗(2)−	vf∗(2)−	NOUN
ejpam-5887	298	2	1	1	NUM
ejpam-5887	298	3	=	=	SYM
ejpam-5887	298	4	vf∗(3	vf∗(3	NOUN
ejpam-5887	298	5	)	)	PUNCT
ejpam-5887	298	6	=	=	SYM
ejpam-5887	299	1	1	1	X
ejpam-5887	299	2	.	.	PUNCT
ejpam-5887	299	3	hence	hence	ADV
ejpam-5887	299	4	,	,	PUNCT
ejpam-5887	299	5	s′(k1,2	s′(k1,2	ADJ
ejpam-5887	299	6	)	)	PUNCT
ejpam-5887	299	7	is	be	AUX
ejpam-5887	299	8	an	an	DET
ejpam-5887	299	9	edge	edge	NOUN
ejpam-5887	299	10	4	4	NUM
ejpam-5887	299	11	-	-	PUNCT
ejpam-5887	299	12	product	product	NOUN
ejpam-5887	299	13	cordial	cordial	ADJ
ejpam-5887	299	14	graph	graph	NOUN
ejpam-5887	299	15	.	.	PUNCT
ejpam-5887	300	1	for	for	ADP
ejpam-5887	300	2	n	n	NOUN
ejpam-5887	300	3	=	=	SYM
ejpam-5887	300	4	1	1	NUM
ejpam-5887	300	5	,	,	PUNCT
ejpam-5887	300	6	|v	|v	ADV
ejpam-5887	300	7	|	|	NOUN
ejpam-5887	300	8	=	=	SYM
ejpam-5887	300	9	4	4	NUM
ejpam-5887	300	10	and	and	CCONJ
ejpam-5887	300	11	|e|	|e|	NOUN
ejpam-5887	300	12	=	=	SYM
ejpam-5887	300	13	3	3	X
ejpam-5887	300	14	.	.	PUNCT
ejpam-5887	301	1	if	if	SCONJ
ejpam-5887	301	2	f	f	PROPN
ejpam-5887	301	3	is	be	AUX
ejpam-5887	301	4	an	an	DET
ejpam-5887	301	5	edge	edge	NOUN
ejpam-5887	301	6	4	4	NUM
ejpam-5887	301	7	-	-	PUNCT
ejpam-5887	301	8	product	product	NOUN
ejpam-5887	301	9	cordial	cordial	ADJ
ejpam-5887	301	10	labeling	labeling	NOUN
ejpam-5887	301	11	of	of	ADP
ejpam-5887	301	12	s′(k1,1	s′(k1,1	NOUN
ejpam-5887	301	13	)	)	PUNCT
ejpam-5887	301	14	,	,	PUNCT
ejpam-5887	301	15	then	then	ADV
ejpam-5887	301	16	ef	ef	PROPN
ejpam-5887	301	17	(	(	PUNCT
ejpam-5887	301	18	i	i	NOUN
ejpam-5887	301	19	)	)	PUNCT
ejpam-5887	301	20	is	be	AUX
ejpam-5887	301	21	either	either	CCONJ
ejpam-5887	301	22	0	0	NUM
ejpam-5887	301	23	or	or	CCONJ
ejpam-5887	301	24	1	1	NUM
ejpam-5887	301	25	for	for	ADP
ejpam-5887	301	26	i	i	PRON
ejpam-5887	301	27	=	=	SYM
ejpam-5887	301	28	0	0	NUM
ejpam-5887	301	29	,	,	PUNCT
ejpam-5887	301	30	1	1	NUM
ejpam-5887	301	31	,	,	PUNCT
ejpam-5887	301	32	2	2	NUM
ejpam-5887	301	33	,	,	PUNCT
ejpam-5887	301	34	3	3	NUM
ejpam-5887	301	35	and	and	CCONJ
ejpam-5887	301	36	vf∗(i	vf∗(i	ADJ
ejpam-5887	301	37	)	)	PUNCT
ejpam-5887	301	38	=	=	SYM
ejpam-5887	301	39	1	1	NUM
ejpam-5887	301	40	for	for	ADP
ejpam-5887	301	41	all	all	DET
ejpam-5887	301	42	i	i	PRON
ejpam-5887	301	43	=	=	NOUN
ejpam-5887	301	44	0	0	NUM
ejpam-5887	301	45	,	,	PUNCT
ejpam-5887	301	46	1	1	NUM
ejpam-5887	301	47	,	,	PUNCT
ejpam-5887	301	48	2	2	NUM
ejpam-5887	301	49	,	,	PUNCT
ejpam-5887	301	50	3	3	NUM
ejpam-5887	301	51	.	.	PUNCT
ejpam-5887	302	1	clearly	clearly	ADV
ejpam-5887	302	2	,	,	PUNCT
ejpam-5887	302	3	ef	ef	PROPN
ejpam-5887	302	4	(	(	PUNCT
ejpam-5887	302	5	0	0	NUM
ejpam-5887	302	6	)	)	PUNCT
ejpam-5887	302	7	=	=	SYM
ejpam-5887	302	8	0	0	NUM
ejpam-5887	302	9	otherwise	otherwise	ADV
ejpam-5887	302	10	vf∗(0	vf∗(0	NOUN
ejpam-5887	302	11	)	)	PUNCT
ejpam-5887	302	12	=	=	SYM
ejpam-5887	302	13	2	2	X
ejpam-5887	302	14	.	.	PUNCT
ejpam-5887	303	1	thus	thus	ADV
ejpam-5887	303	2	,	,	PUNCT
ejpam-5887	303	3	ef	ef	PROPN
ejpam-5887	303	4	(	(	PUNCT
ejpam-5887	303	5	i	i	NOUN
ejpam-5887	303	6	)	)	PUNCT
ejpam-5887	303	7	=	=	SYM
ejpam-5887	303	8	1	1	NUM
ejpam-5887	303	9	for	for	ADP
ejpam-5887	303	10	all	all	DET
ejpam-5887	303	11	i	i	PRON
ejpam-5887	303	12	=	=	NOUN
ejpam-5887	303	13	1	1	NUM
ejpam-5887	303	14	,	,	PUNCT
ejpam-5887	303	15	2	2	NUM
ejpam-5887	303	16	,	,	PUNCT
ejpam-5887	303	17	3	3	NUM
ejpam-5887	303	18	.	.	PUNCT
ejpam-5887	303	19	but	but	CCONJ
ejpam-5887	303	20	ef	ef	PROPN
ejpam-5887	303	21	(	(	PUNCT
ejpam-5887	303	22	2	2	NUM
ejpam-5887	303	23	)	)	PUNCT
ejpam-5887	303	24	=	=	SYM
ejpam-5887	303	25	1	1	NUM
ejpam-5887	303	26	implies	imply	VERB
ejpam-5887	303	27	vf∗(0	vf∗(0	NOUN
ejpam-5887	303	28	)	)	PUNCT
ejpam-5887	303	29	=	=	SYM
ejpam-5887	303	30	0	0	NUM
ejpam-5887	303	31	and	and	CCONJ
ejpam-5887	303	32	vf∗(2	vf∗(2	ADJ
ejpam-5887	303	33	)	)	PUNCT
ejpam-5887	303	34	=	=	SYM
ejpam-5887	303	35	2	2	X
ejpam-5887	303	36	.	.	X
ejpam-5887	303	37	therefore	therefore	ADV
ejpam-5887	303	38	,	,	PUNCT
ejpam-5887	303	39	|vf∗(0)−	|vf∗(0)−	PROPN
ejpam-5887	303	40	vf∗(2)|	vf∗(2)|	NOUN
ejpam-5887	303	41	>	>	X
ejpam-5887	303	42	1	1	NUM
ejpam-5887	303	43	which	which	PRON
ejpam-5887	303	44	is	be	AUX
ejpam-5887	303	45	a	a	DET
ejpam-5887	303	46	contradiction	contradiction	NOUN
ejpam-5887	303	47	.	.	PUNCT
ejpam-5887	304	1	hence	hence	ADV
ejpam-5887	304	2	,	,	PUNCT
ejpam-5887	304	3	s′(k1,1	s′(k1,1	CCONJ
ejpam-5887	304	4	)	)	PUNCT
ejpam-5887	304	5	is	be	AUX
ejpam-5887	304	6	not	not	PART
ejpam-5887	304	7	an	an	DET
ejpam-5887	304	8	edge	edge	NOUN
ejpam-5887	304	9	4	4	NUM
ejpam-5887	304	10	-	-	PUNCT
ejpam-5887	304	11	product	product	NOUN
ejpam-5887	304	12	cordial	cordial	ADJ
ejpam-5887	304	13	graph	graph	NOUN
ejpam-5887	304	14	.	.	PUNCT
ejpam-5887	305	1	for	for	ADP
ejpam-5887	305	2	n	n	NOUN
ejpam-5887	305	3	=	=	SYM
ejpam-5887	305	4	3	3	NUM
ejpam-5887	305	5	,	,	PUNCT
ejpam-5887	305	6	|v	|v	ADV
ejpam-5887	305	7	|	|	NOUN
ejpam-5887	305	8	=	=	SYM
ejpam-5887	305	9	8	8	NUM
ejpam-5887	305	10	and	and	CCONJ
ejpam-5887	305	11	|e|	|e|	NOUN
ejpam-5887	305	12	=	=	SYM
ejpam-5887	305	13	9	9	X
ejpam-5887	305	14	.	.	PUNCT
ejpam-5887	305	15	by	by	ADP
ejpam-5887	305	16	theorem	theorem	NOUN
ejpam-5887	305	17	5	5	NUM
ejpam-5887	305	18	,	,	PUNCT
ejpam-5887	305	19	s′(k1,3	s′(k1,3	PRON
ejpam-5887	305	20	)	)	PUNCT
ejpam-5887	305	21	is	be	AUX
ejpam-5887	305	22	not	not	PART
ejpam-5887	305	23	an	an	DET
ejpam-5887	305	24	edge	edge	NOUN
ejpam-5887	305	25	4	4	NUM
ejpam-5887	305	26	-	-	PUNCT
ejpam-5887	305	27	product	product	NOUN
ejpam-5887	305	28	cordial	cordial	ADJ
ejpam-5887	305	29	graph	graph	NOUN
ejpam-5887	305	30	.	.	PUNCT
ejpam-5887	306	1	also	also	ADV
ejpam-5887	306	2	,	,	PUNCT
ejpam-5887	306	3	by	by	ADP
ejpam-5887	306	4	theorem	theorem	NOUN
ejpam-5887	306	5	10	10	NUM
ejpam-5887	306	6	,	,	PUNCT
ejpam-5887	306	7	s′(k1,n	s′(k1,n	X
ejpam-5887	306	8	)	)	PUNCT
ejpam-5887	306	9	is	be	AUX
ejpam-5887	306	10	not	not	PART
ejpam-5887	306	11	an	an	DET
ejpam-5887	306	12	edge	edge	NOUN
ejpam-5887	306	13	4	4	NUM
ejpam-5887	306	14	-	-	PUNCT
ejpam-5887	306	15	product	product	NOUN
ejpam-5887	306	16	cordial	cordial	ADJ
ejpam-5887	306	17	graph	graph	NOUN
ejpam-5887	306	18	if	if	SCONJ
ejpam-5887	306	19	n	n	NUM
ejpam-5887	306	20	≥	≥	NOUN
ejpam-5887	306	21	4	4	NUM
ejpam-5887	306	22	.	.	PUNCT
ejpam-5887	306	23	theorem	theorem	VERB
ejpam-5887	306	24	13	13	NUM
ejpam-5887	306	25	.	.	PUNCT
ejpam-5887	307	1	the	the	DET
ejpam-5887	307	2	graph	graph	NOUN
ejpam-5887	307	3	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	307	4	)	)	PUNCT
ejpam-5887	307	5	admits	admit	VERB
ejpam-5887	307	6	an	an	DET
ejpam-5887	307	7	edge	edge	NOUN
ejpam-5887	307	8	5	5	NUM
ejpam-5887	307	9	-	-	PUNCT
ejpam-5887	307	10	product	product	NOUN
ejpam-5887	307	11	cordial	cordial	ADJ
ejpam-5887	307	12	labeling	labeling	NOUN
ejpam-5887	307	13	if	if	SCONJ
ejpam-5887	307	14	and	and	CCONJ
ejpam-5887	307	15	only	only	ADV
ejpam-5887	307	16	if	if	SCONJ
ejpam-5887	307	17	n	n	PRON
ejpam-5887	307	18	≤	≤	ADV
ejpam-5887	307	19	3	3	NUM
ejpam-5887	307	20	.	.	PUNCT
ejpam-5887	308	1	proof	proof	NOUN
ejpam-5887	308	2	.	.	PUNCT
ejpam-5887	309	1	let	let	VERB
ejpam-5887	309	2	the	the	DET
ejpam-5887	309	3	vertex	vertex	NOUN
ejpam-5887	309	4	set	set	NOUN
ejpam-5887	309	5	and	and	CCONJ
ejpam-5887	309	6	edge	edge	NOUN
ejpam-5887	309	7	set	set	NOUN
ejpam-5887	309	8	of	of	ADP
ejpam-5887	309	9	s′(k1,n	s′(k1,n	ADJ
ejpam-5887	309	10	)	)	PUNCT
ejpam-5887	309	11	be	be	AUX
ejpam-5887	309	12	v	v	ADP
ejpam-5887	309	13	(	(	PUNCT
ejpam-5887	309	14	s′(k1,n	s′(k1,n	NOUN
ejpam-5887	309	15	)	)	PUNCT
ejpam-5887	309	16	)	)	PUNCT
ejpam-5887	310	1	=	=	PRON
ejpam-5887	310	2	{	{	PUNCT
ejpam-5887	310	3	u	u	PROPN
ejpam-5887	310	4	,	,	PUNCT
ejpam-5887	310	5	ui	ui	PROPN
ejpam-5887	310	6	,	,	PUNCT
ejpam-5887	310	7	v	v	NOUN
ejpam-5887	310	8	,	,	PUNCT
ejpam-5887	310	9	vi	vi	NOUN
ejpam-5887	310	10	;	;	PUNCT
ejpam-5887	310	11	1	1	NUM
ejpam-5887	310	12	≤	≤	NUM
ejpam-5887	310	13	i	i	PRON
ejpam-5887	310	14	≤	≤	NOUN
ejpam-5887	310	15	n	n	CCONJ
ejpam-5887	310	16	}	}	PUNCT
ejpam-5887	310	17	and	and	CCONJ
ejpam-5887	310	18	e(s′(k1,n	e(s′(k1,n	NOUN
ejpam-5887	310	19	)	)	PUNCT
ejpam-5887	310	20	)	)	PUNCT
ejpam-5887	311	1	=	=	PRON
ejpam-5887	311	2	{	{	PUNCT
ejpam-5887	311	3	uui	uui	PROPN
ejpam-5887	311	4	,	,	PUNCT
ejpam-5887	311	5	vui	vui	PROPN
ejpam-5887	311	6	,	,	PUNCT
ejpam-5887	311	7	uvi	uvi	PROPN
ejpam-5887	311	8	;	;	PUNCT
ejpam-5887	311	9	1	1	NUM
ejpam-5887	311	10	≤	≤	NUM
ejpam-5887	311	11	i	i	PRON
ejpam-5887	311	12	≤	≤	NOUN
ejpam-5887	311	13	n	n	CCONJ
ejpam-5887	311	14	}	}	PUNCT
ejpam-5887	311	15	respectively	respectively	ADV
ejpam-5887	311	16	.	.	PUNCT
ejpam-5887	312	1	define	define	VERB
ejpam-5887	312	2	an	an	DET
ejpam-5887	312	3	edge	edge	NOUN
ejpam-5887	312	4	labeling	labeling	NOUN
ejpam-5887	312	5	f	f	NOUN
ejpam-5887	312	6	:	:	PUNCT
ejpam-5887	313	1	s′(k1,n	s′(k1,n	X
ejpam-5887	313	2	)	)	PUNCT
ejpam-5887	313	3	→	→	SYM
ejpam-5887	313	4	{	{	PUNCT
ejpam-5887	313	5	0	0	NUM
ejpam-5887	313	6	,	,	PUNCT
ejpam-5887	313	7	1	1	NUM
ejpam-5887	313	8	,	,	PUNCT
ejpam-5887	313	9	...	...	PUNCT
ejpam-5887	313	10	,	,	PUNCT
ejpam-5887	313	11	k	k	PROPN
ejpam-5887	313	12	−	−	PROPN
ejpam-5887	313	13	1	1	NUM
ejpam-5887	313	14	}	}	PUNCT
ejpam-5887	313	15	for	for	ADP
ejpam-5887	313	16	n	n	DET
ejpam-5887	313	17	≤	≤	ADV
ejpam-5887	313	18	3	3	NUM
ejpam-5887	313	19	as	as	SCONJ
ejpam-5887	313	20	follows	follow	VERB
ejpam-5887	313	21	:	:	PUNCT
ejpam-5887	313	22	f(uui	f(uui	ADJ
ejpam-5887	313	23	)	)	PUNCT
ejpam-5887	313	24	=	=	PUNCT
ejpam-5887	314	1			X
ejpam-5887	314	2	0	0	NUM
ejpam-5887	314	3	;	;	PUNCT
ejpam-5887	314	4	i	i	PRON
ejpam-5887	314	5	=	=	NOUN
ejpam-5887	314	6	1	1	NUM
ejpam-5887	314	7	,	,	PUNCT
ejpam-5887	314	8	n	n	NOUN
ejpam-5887	314	9	=	=	SYM
ejpam-5887	314	10	2	2	NUM
ejpam-5887	314	11	,	,	PUNCT
ejpam-5887	314	12	3	3	NUM
ejpam-5887	314	13	1	1	NUM
ejpam-5887	314	14	;	;	PUNCT
ejpam-5887	314	15	i	i	PRON
ejpam-5887	314	16	=	=	SYM
ejpam-5887	314	17	2	2	NUM
ejpam-5887	314	18	,	,	PUNCT
ejpam-5887	314	19	n	n	NOUN
ejpam-5887	314	20	=	=	SYM
ejpam-5887	314	21	2	2	NUM
ejpam-5887	314	22	,	,	PUNCT
ejpam-5887	314	23	3	3	NUM
ejpam-5887	314	24	2	2	NUM
ejpam-5887	314	25	;	;	PUNCT
ejpam-5887	314	26	i	i	PRON
ejpam-5887	314	27	=	=	SYM
ejpam-5887	314	28	n	n	X
ejpam-5887	314	29	,	,	PUNCT
ejpam-5887	314	30	n	n	NOUN
ejpam-5887	314	31	=	=	SYM
ejpam-5887	314	32	1	1	NUM
ejpam-5887	314	33	,	,	PUNCT
ejpam-5887	314	34	3	3	NUM
ejpam-5887	314	35	,	,	PUNCT
ejpam-5887	314	36	f(vui	f(vui	NOUN
ejpam-5887	314	37	)	)	PUNCT
ejpam-5887	314	38	=	=	PUNCT
ejpam-5887	315	1			PUNCT
ejpam-5887	315	2	1	1	NUM
ejpam-5887	315	3	;	;	PUNCT
ejpam-5887	315	4	i	i	PRON
ejpam-5887	315	5	=	=	NOUN
ejpam-5887	315	6	1	1	NUM
ejpam-5887	315	7	,	,	PUNCT
ejpam-5887	315	8	n	n	NOUN
ejpam-5887	315	9	=	=	SYM
ejpam-5887	315	10	3	3	NUM
ejpam-5887	315	11	;	;	PUNCT
ejpam-5887	315	12	i	i	PRON
ejpam-5887	315	13	=	=	SYM
ejpam-5887	315	14	n	n	NOUN
ejpam-5887	315	15	=	=	SYM
ejpam-5887	315	16	2	2	NUM
ejpam-5887	315	17	3	3	NUM
ejpam-5887	315	18	;	;	PUNCT
ejpam-5887	315	19	i	i	PRON
ejpam-5887	315	20	=	=	SYM
ejpam-5887	315	21	n	n	NOUN
ejpam-5887	315	22	=	=	SYM
ejpam-5887	315	23	3	3	NUM
ejpam-5887	315	24	4	4	NUM
ejpam-5887	315	25	;	;	PUNCT
ejpam-5887	315	26	i	i	PRON
ejpam-5887	315	27	=	=	NOUN
ejpam-5887	315	28	1	1	NUM
ejpam-5887	315	29	,	,	PUNCT
ejpam-5887	315	30	n	n	NOUN
ejpam-5887	315	31	=	=	SYM
ejpam-5887	315	32	1	1	NUM
ejpam-5887	315	33	,	,	PUNCT
ejpam-5887	315	34	2	2	NUM
ejpam-5887	315	35	;	;	PUNCT
ejpam-5887	315	36	i	i	NOUN
ejpam-5887	315	37	=	=	SYM
ejpam-5887	315	38	2	2	NUM
ejpam-5887	315	39	,	,	PUNCT
ejpam-5887	315	40	n	n	NOUN
ejpam-5887	315	41	=	=	SYM
ejpam-5887	315	42	3	3	NUM
ejpam-5887	315	43	,	,	PUNCT
ejpam-5887	315	44	f(uvi	f(uvi	NOUN
ejpam-5887	315	45	)	)	PUNCT
ejpam-5887	315	46	=	=	SYM
ejpam-5887	315	47			NUM
ejpam-5887	315	48	1	1	NUM
ejpam-5887	315	49	;	;	PUNCT
ejpam-5887	315	50	i	i	PRON
ejpam-5887	315	51	=	=	SYM
ejpam-5887	315	52	n	n	NOUN
ejpam-5887	315	53	=	=	SYM
ejpam-5887	315	54	1	1	NUM
ejpam-5887	315	55	2	2	NUM
ejpam-5887	315	56	;	;	PUNCT
ejpam-5887	315	57	i	i	NOUN
ejpam-5887	315	58	=	=	NOUN
ejpam-5887	315	59	1	1	NUM
ejpam-5887	315	60	,	,	PUNCT
ejpam-5887	315	61	n	n	NOUN
ejpam-5887	315	62	=	=	SYM
ejpam-5887	315	63	2	2	NUM
ejpam-5887	315	64	,	,	PUNCT
ejpam-5887	315	65	3	3	NUM
ejpam-5887	315	66	3	3	NUM
ejpam-5887	315	67	;	;	PUNCT
ejpam-5887	315	68	i	i	PRON
ejpam-5887	315	69	=	=	SYM
ejpam-5887	315	70	2	2	NUM
ejpam-5887	315	71	,	,	PUNCT
ejpam-5887	315	72	n	n	NOUN
ejpam-5887	315	73	=	=	SYM
ejpam-5887	315	74	2	2	NUM
ejpam-5887	315	75	,	,	PUNCT
ejpam-5887	315	76	3	3	NUM
ejpam-5887	315	77	4	4	NUM
ejpam-5887	315	78	;	;	PUNCT
ejpam-5887	315	79	i	i	PRON
ejpam-5887	315	80	=	=	SYM
ejpam-5887	315	81	n	n	PROPN
ejpam-5887	315	82	=	=	SYM
ejpam-5887	315	83	3	3	X
ejpam-5887	315	84	.	.	PUNCT
ejpam-5887	315	85	clearly	clearly	ADV
ejpam-5887	315	86	,	,	PUNCT
ejpam-5887	315	87	|ef	|ef	PROPN
ejpam-5887	315	88	(	(	PUNCT
ejpam-5887	315	89	i)−ef	i)−ef	X
ejpam-5887	315	90	(	(	PUNCT
ejpam-5887	315	91	j)|	j)|	NOUN
ejpam-5887	315	92	≤	≤	NUM
ejpam-5887	315	93	1	1	NUM
ejpam-5887	315	94	and	and	CCONJ
ejpam-5887	315	95	|vf∗(i)−vf∗(j)|	|vf∗(i)−vf∗(j)|	NOUN
ejpam-5887	315	96	≤	≤	NUM
ejpam-5887	315	97	1	1	NUM
ejpam-5887	315	98	for	for	ADP
ejpam-5887	315	99	i	i	PRON
ejpam-5887	315	100	,	,	PUNCT
ejpam-5887	315	101	j	j	PROPN
ejpam-5887	315	102	∈	∈	PROPN
ejpam-5887	315	103	{	{	PUNCT
ejpam-5887	315	104	0	0	NUM
ejpam-5887	315	105	,	,	PUNCT
ejpam-5887	315	106	1	1	NUM
ejpam-5887	315	107	,	,	PUNCT
ejpam-5887	315	108	2	2	NUM
ejpam-5887	315	109	,	,	PUNCT
ejpam-5887	315	110	3	3	NUM
ejpam-5887	315	111	,	,	PUNCT
ejpam-5887	315	112	4	4	NUM
ejpam-5887	315	113	}	}	PUNCT
ejpam-5887	315	114	.	.	PUNCT
ejpam-5887	316	1	hence	hence	ADV
ejpam-5887	316	2	,	,	PUNCT
ejpam-5887	316	3	s′(k1,n	s′(k1,n	X
ejpam-5887	316	4	)	)	PUNCT
ejpam-5887	316	5	is	be	AUX
ejpam-5887	316	6	an	an	DET
ejpam-5887	316	7	edge	edge	NOUN
ejpam-5887	316	8	5	5	NUM
ejpam-5887	316	9	-	-	PUNCT
ejpam-5887	316	10	product	product	NOUN
ejpam-5887	316	11	cordial	cordial	ADJ
ejpam-5887	316	12	graph	graph	NOUN
ejpam-5887	316	13	if	if	SCONJ
ejpam-5887	316	14	n	n	ADV
ejpam-5887	316	15	≤	≤	ADV
ejpam-5887	316	16	3	3	NUM
ejpam-5887	316	17	.	.	PUNCT
ejpam-5887	316	18	n.	n.	PROPN
ejpam-5887	316	19	m.	m.	PROPN
ejpam-5887	316	20	noureldeen	noureldeen	INTJ
ejpam-5887	316	21	et	et	PROPN
ejpam-5887	316	22	al	al	PROPN
ejpam-5887	316	23	.	.	PUNCT
ejpam-5887	316	24	/	/	SYM
ejpam-5887	316	25	eur	eur	PROPN
ejpam-5887	316	26	.	.	PUNCT
ejpam-5887	317	1	j.	j.	PROPN
ejpam-5887	317	2	pure	pure	PROPN
ejpam-5887	317	3	appl	appl	PROPN
ejpam-5887	317	4	.	.	PROPN
ejpam-5887	317	5	math	math	PROPN
ejpam-5887	317	6	,	,	PUNCT
ejpam-5887	317	7	18	18	NUM
ejpam-5887	317	8	(	(	PUNCT
ejpam-5887	317	9	2	2	NUM
ejpam-5887	317	10	)	)	PUNCT
ejpam-5887	317	11	(	(	PUNCT
ejpam-5887	317	12	2025	2025	NUM
ejpam-5887	317	13	)	)	PUNCT
ejpam-5887	317	14	,	,	PUNCT
ejpam-5887	317	15	5887	5887	NUM
ejpam-5887	317	16	10	10	NUM
ejpam-5887	317	17	of	of	ADP
ejpam-5887	317	18	21	21	NUM
ejpam-5887	317	19	for	for	ADP
ejpam-5887	317	20	n	n	NOUN
ejpam-5887	317	21	=	=	SYM
ejpam-5887	317	22	4	4	NUM
ejpam-5887	317	23	,	,	PUNCT
ejpam-5887	317	24	|v	|v	ADV
ejpam-5887	317	25	|	|	NOUN
ejpam-5887	317	26	=	=	NOUN
ejpam-5887	317	27	10	10	NUM
ejpam-5887	317	28	and	and	CCONJ
ejpam-5887	317	29	|e|	|e|	NOUN
ejpam-5887	317	30	=	=	NOUN
ejpam-5887	317	31	12	12	NUM
ejpam-5887	317	32	.	.	PUNCT
ejpam-5887	318	1	by	by	ADP
ejpam-5887	318	2	theorem	theorem	NOUN
ejpam-5887	318	3	5	5	NUM
ejpam-5887	318	4	,	,	PUNCT
ejpam-5887	318	5	s′(k1,4	s′(k1,4	ADV
ejpam-5887	318	6	)	)	PUNCT
ejpam-5887	318	7	is	be	AUX
ejpam-5887	318	8	not	not	PART
ejpam-5887	318	9	an	an	DET
ejpam-5887	318	10	edge	edge	NOUN
ejpam-5887	318	11	5	5	NUM
ejpam-5887	318	12	-	-	PUNCT
ejpam-5887	318	13	product	product	NOUN
ejpam-5887	318	14	cordial	cordial	ADJ
ejpam-5887	318	15	graph	graph	NOUN
ejpam-5887	318	16	.	.	PUNCT
ejpam-5887	319	1	also	also	ADV
ejpam-5887	319	2	,	,	PUNCT
ejpam-5887	319	3	by	by	ADP
ejpam-5887	319	4	theorem	theorem	NOUN
ejpam-5887	319	5	10	10	NUM
ejpam-5887	319	6	,	,	PUNCT
ejpam-5887	319	7	s′(k1,n	s′(k1,n	X
ejpam-5887	319	8	)	)	PUNCT
ejpam-5887	319	9	is	be	AUX
ejpam-5887	319	10	not	not	PART
ejpam-5887	319	11	an	an	DET
ejpam-5887	319	12	edge	edge	NOUN
ejpam-5887	319	13	5	5	NUM
ejpam-5887	319	14	-	-	PUNCT
ejpam-5887	319	15	product	product	NOUN
ejpam-5887	319	16	cordial	cordial	ADJ
ejpam-5887	319	17	graph	graph	NOUN
ejpam-5887	319	18	if	if	SCONJ
ejpam-5887	319	19	n	n	NUM
ejpam-5887	319	20	≥	≥	NOUN
ejpam-5887	319	21	5	5	NUM
ejpam-5887	319	22	.	.	NOUN
ejpam-5887	319	23	4	4	NUM
ejpam-5887	319	24	.	.	X
ejpam-5887	319	25	edge	edge	NOUN
ejpam-5887	319	26	k	k	NOUN
ejpam-5887	319	27	-	-	PUNCT
ejpam-5887	319	28	product	product	NOUN
ejpam-5887	319	29	cordial	cordial	ADJ
ejpam-5887	319	30	labeling	labeling	NOUN
ejpam-5887	319	31	of	of	ADP
ejpam-5887	319	32	path	path	NOUN
ejpam-5887	319	33	union	union	NOUN
ejpam-5887	319	34	of	of	ADP
ejpam-5887	319	35	graphs	graph	NOUN
ejpam-5887	319	36	in	in	ADP
ejpam-5887	319	37	this	this	DET
ejpam-5887	319	38	section	section	NOUN
ejpam-5887	319	39	,	,	PUNCT
ejpam-5887	319	40	we	we	PRON
ejpam-5887	319	41	explore	explore	VERB
ejpam-5887	319	42	the	the	DET
ejpam-5887	319	43	edge	edge	NOUN
ejpam-5887	319	44	k	k	NOUN
ejpam-5887	319	45	-	-	PUNCT
ejpam-5887	319	46	product	product	NOUN
ejpam-5887	319	47	cordial	cordial	ADJ
ejpam-5887	319	48	behavior	behavior	NOUN
ejpam-5887	319	49	of	of	ADP
ejpam-5887	319	50	the	the	DET
ejpam-5887	319	51	path	path	NOUN
ejpam-5887	319	52	union	union	PROPN
ejpam-5887	319	53	of	of	ADP
ejpam-5887	319	54	star	star	PROPN
ejpam-5887	319	55	,	,	PUNCT
ejpam-5887	319	56	bistar	bistar	NOUN
ejpam-5887	319	57	and	and	CCONJ
ejpam-5887	319	58	cycle	cycle	NOUN
ejpam-5887	319	59	graphs	graph	NOUN
ejpam-5887	319	60	.	.	PUNCT
ejpam-5887	320	1	in	in	ADP
ejpam-5887	320	2	the	the	DET
ejpam-5887	320	3	following	follow	VERB
ejpam-5887	320	4	general	general	ADJ
ejpam-5887	320	5	result	result	NOUN
ejpam-5887	320	6	,	,	PUNCT
ejpam-5887	320	7	we	we	PRON
ejpam-5887	320	8	show	show	VERB
ejpam-5887	320	9	that	that	SCONJ
ejpam-5887	320	10	the	the	DET
ejpam-5887	320	11	path	path	NOUN
ejpam-5887	320	12	union	union	NOUN
ejpam-5887	320	13	of	of	ADP
ejpam-5887	320	14	an	an	DET
ejpam-5887	320	15	edge	edge	NOUN
ejpam-5887	320	16	k	k	NOUN
ejpam-5887	320	17	-	-	PUNCT
ejpam-5887	320	18	product	product	NOUN
ejpam-5887	320	19	cordial	cordial	ADJ
ejpam-5887	320	20	graph	graph	NOUN
ejpam-5887	320	21	with	with	ADP
ejpam-5887	320	22	multiple	multiple	NOUN
ejpam-5887	320	23	of	of	ADP
ejpam-5887	320	24	k	k	PROPN
ejpam-5887	320	25	edges	edge	NOUN
ejpam-5887	320	26	also	also	ADV
ejpam-5887	320	27	admits	admit	VERB
ejpam-5887	320	28	an	an	DET
ejpam-5887	320	29	edge	edge	NOUN
ejpam-5887	320	30	k	k	NOUN
ejpam-5887	320	31	-	-	PUNCT
ejpam-5887	320	32	product	product	NOUN
ejpam-5887	320	33	cordial	cordial	ADJ
ejpam-5887	320	34	labeling	labeling	NOUN
ejpam-5887	320	35	.	.	PUNCT
ejpam-5887	321	1	theorem	theorem	VERB
ejpam-5887	321	2	14	14	NUM
ejpam-5887	321	3	.	.	PUNCT
ejpam-5887	322	1	let	let	VERB
ejpam-5887	322	2	g	g	PRON
ejpam-5887	322	3	be	be	AUX
ejpam-5887	322	4	an	an	DET
ejpam-5887	322	5	edge	edge	NOUN
ejpam-5887	322	6	k	k	NOUN
ejpam-5887	322	7	-	-	PUNCT
ejpam-5887	322	8	product	product	NOUN
ejpam-5887	322	9	cordial	cordial	ADJ
ejpam-5887	322	10	graph	graph	NOUN
ejpam-5887	322	11	with	with	ADP
ejpam-5887	322	12	multiple	multiple	NOUN
ejpam-5887	322	13	of	of	ADP
ejpam-5887	322	14	k	k	PROPN
ejpam-5887	322	15	edges	edge	NOUN
ejpam-5887	322	16	.	.	PUNCT
ejpam-5887	323	1	then	then	ADV
ejpam-5887	323	2	p	p	X
ejpam-5887	323	3	(	(	PUNCT
ejpam-5887	323	4	n.gv	n.gv	PROPN
ejpam-5887	323	5	)	)	PUNCT
ejpam-5887	323	6	,	,	PUNCT
ejpam-5887	323	7	where	where	SCONJ
ejpam-5887	323	8	v	v	NOUN
ejpam-5887	323	9	is	be	AUX
ejpam-5887	323	10	a	a	DET
ejpam-5887	323	11	vertex	vertex	NOUN
ejpam-5887	323	12	of	of	ADP
ejpam-5887	323	13	g	g	NOUN
ejpam-5887	323	14	such	such	ADJ
ejpam-5887	323	15	that	that	SCONJ
ejpam-5887	323	16	at	at	ADV
ejpam-5887	323	17	least	least	ADJ
ejpam-5887	323	18	one	one	NUM
ejpam-5887	323	19	of	of	ADP
ejpam-5887	323	20	its	its	PRON
ejpam-5887	323	21	incident	incident	NOUN
ejpam-5887	323	22	edges	edge	NOUN
ejpam-5887	323	23	is	be	AUX
ejpam-5887	323	24	labeled	label	VERB
ejpam-5887	323	25	with	with	ADP
ejpam-5887	323	26	0	0	NUM
ejpam-5887	323	27	admits	admit	VERB
ejpam-5887	323	28	an	an	DET
ejpam-5887	323	29	edge	edge	NOUN
ejpam-5887	323	30	k	k	NOUN
ejpam-5887	323	31	-	-	PUNCT
ejpam-5887	323	32	product	product	NOUN
ejpam-5887	323	33	cordial	cordial	ADJ
ejpam-5887	323	34	labeling	labeling	NOUN
ejpam-5887	323	35	.	.	PUNCT
ejpam-5887	324	1	proof	proof	NOUN
ejpam-5887	324	2	.	.	PUNCT
ejpam-5887	325	1	let	let	VERB
ejpam-5887	325	2	the	the	DET
ejpam-5887	325	3	vertex	vertex	NOUN
ejpam-5887	325	4	and	and	CCONJ
ejpam-5887	325	5	edge	edge	NOUN
ejpam-5887	325	6	set	set	NOUN
ejpam-5887	325	7	of	of	ADP
ejpam-5887	325	8	p	p	PROPN
ejpam-5887	325	9	(	(	PUNCT
ejpam-5887	325	10	n.gv	n.gv	NOUN
ejpam-5887	325	11	)	)	PUNCT
ejpam-5887	325	12	be	be	VERB
ejpam-5887	325	13	v	v	ADP
ejpam-5887	325	14	(	(	PUNCT
ejpam-5887	325	15	p	p	X
ejpam-5887	325	16	(	(	PUNCT
ejpam-5887	325	17	n.gv	n.gv	NOUN
ejpam-5887	325	18	)	)	PUNCT
ejpam-5887	325	19	)	)	PUNCT
ejpam-5887	326	1	=	=	PUNCT
ejpam-5887	326	2	⋃	⋃	ADP
ejpam-5887	326	3	1≤i≤n	1≤i≤n	NUM
ejpam-5887	326	4	v	v	NOUN
ejpam-5887	326	5	(	(	PUNCT
ejpam-5887	326	6	gi	gi	NOUN
ejpam-5887	326	7	)	)	PUNCT
ejpam-5887	326	8	and	and	CCONJ
ejpam-5887	326	9	e(p	e(p	PROPN
ejpam-5887	326	10	(	(	PUNCT
ejpam-5887	326	11	n.gv	n.gv	NOUN
ejpam-5887	326	12	)	)	PUNCT
ejpam-5887	326	13	)	)	PUNCT
ejpam-5887	327	1	=	=	PUNCT
ejpam-5887	328	1	⋃	⋃	NOUN
ejpam-5887	328	2	1≤i≤ne(gi)∪{ei	1≤i≤ne(gi)∪{ei	NUM
ejpam-5887	328	3	:	:	PUNCT
ejpam-5887	328	4	ei	ei	X
ejpam-5887	328	5	=	=	PUNCT
ejpam-5887	328	6	vivi+1	vivi+1	PROPN
ejpam-5887	328	7	,	,	PUNCT
ejpam-5887	328	8	vi	vi	PROPN
ejpam-5887	328	9	∈	∈	PROPN
ejpam-5887	328	10	v	v	NOUN
ejpam-5887	328	11	(	(	PUNCT
ejpam-5887	328	12	gi	gi	INTJ
ejpam-5887	328	13	)	)	PUNCT
ejpam-5887	328	14	,	,	PUNCT
ejpam-5887	328	15	1	1	NUM
ejpam-5887	328	16	≤	≤	NUM
ejpam-5887	328	17	i	i	X
ejpam-5887	328	18	≤	≤	NOUN
ejpam-5887	328	19	n−1	n−1	PROPN
ejpam-5887	328	20	}	}	PUNCT
ejpam-5887	328	21	respectively	respectively	ADV
ejpam-5887	328	22	.	.	PUNCT
ejpam-5887	329	1	let	let	VERB
ejpam-5887	329	2	g	g	PRON
ejpam-5887	329	3	be	be	AUX
ejpam-5887	329	4	an	an	DET
ejpam-5887	329	5	edge	edge	NOUN
ejpam-5887	329	6	k	k	NOUN
ejpam-5887	329	7	-	-	PUNCT
ejpam-5887	329	8	product	product	NOUN
ejpam-5887	329	9	cordial	cordial	ADJ
ejpam-5887	329	10	labeling	labeling	NOUN
ejpam-5887	329	11	of	of	ADP
ejpam-5887	329	12	g.	g.	PROPN
ejpam-5887	329	13	since	since	SCONJ
ejpam-5887	329	14	g	g	PROPN
ejpam-5887	329	15	has	have	VERB
ejpam-5887	329	16	kt	kt	PROPN
ejpam-5887	329	17	edges	edge	NOUN
ejpam-5887	329	18	,	,	PUNCT
ejpam-5887	329	19	eg(i	eg(i	PUNCT
ejpam-5887	329	20	)	)	PUNCT
ejpam-5887	330	1	=	=	SYM
ejpam-5887	330	2	t	t	PROPN
ejpam-5887	330	3	for	for	ADP
ejpam-5887	330	4	all	all	DET
ejpam-5887	330	5	i	i	NOUN
ejpam-5887	330	6	=	=	NOUN
ejpam-5887	330	7	0	0	NUM
ejpam-5887	330	8	,	,	PUNCT
ejpam-5887	330	9	1	1	NUM
ejpam-5887	330	10	,	,	PUNCT
ejpam-5887	330	11	...	...	PUNCT
ejpam-5887	330	12	,	,	PUNCT
ejpam-5887	330	13	k	k	PROPN
ejpam-5887	331	1	−	−	PROPN
ejpam-5887	331	2	1	1	NUM
ejpam-5887	331	3	and	and	CCONJ
ejpam-5887	331	4	|vg∗(i)−	|vg∗(i)−	NOUN
ejpam-5887	331	5	vg∗(j)|	vg∗(j)|	X
ejpam-5887	331	6	≤	≤	NOUN
ejpam-5887	331	7	1	1	NUM
ejpam-5887	331	8	for	for	ADP
ejpam-5887	331	9	i	i	PRON
ejpam-5887	331	10	,	,	PUNCT
ejpam-5887	331	11	j	j	PROPN
ejpam-5887	331	12	∈	∈	PROPN
ejpam-5887	331	13	{	{	PUNCT
ejpam-5887	331	14	0	0	NUM
ejpam-5887	331	15	,	,	PUNCT
ejpam-5887	331	16	1	1	NUM
ejpam-5887	331	17	,	,	PUNCT
ejpam-5887	331	18	...	...	PUNCT
ejpam-5887	331	19	,	,	PUNCT
ejpam-5887	331	20	k	k	PROPN
ejpam-5887	331	21	−	−	PROPN
ejpam-5887	331	22	1	1	NUM
ejpam-5887	331	23	}	}	PUNCT
ejpam-5887	331	24	.	.	PUNCT
ejpam-5887	332	1	define	define	VERB
ejpam-5887	332	2	an	an	DET
ejpam-5887	332	3	edge	edge	NOUN
ejpam-5887	332	4	labeling	labeling	NOUN
ejpam-5887	332	5	f	f	NOUN
ejpam-5887	332	6	:	:	PUNCT
ejpam-5887	333	1	e(p	e(p	PROPN
ejpam-5887	333	2	(	(	PUNCT
ejpam-5887	333	3	n.gv	n.gv	NOUN
ejpam-5887	333	4	)	)	PUNCT
ejpam-5887	333	5	)	)	PUNCT
ejpam-5887	333	6	→	→	PUNCT
ejpam-5887	333	7	{	{	PUNCT
ejpam-5887	333	8	0	0	NUM
ejpam-5887	333	9	,	,	PUNCT
ejpam-5887	333	10	1	1	NUM
ejpam-5887	333	11	,	,	PUNCT
ejpam-5887	333	12	...	...	PUNCT
ejpam-5887	333	13	,	,	PUNCT
ejpam-5887	333	14	k−1	k−1	PROPN
ejpam-5887	333	15	}	}	PUNCT
ejpam-5887	333	16	for	for	ADP
ejpam-5887	333	17	n	n	PRON
ejpam-5887	333	18	≡	≡	PROPN
ejpam-5887	333	19	r	r	NOUN
ejpam-5887	333	20	(	(	PUNCT
ejpam-5887	333	21	mod	mod	PROPN
ejpam-5887	333	22	k	k	PROPN
ejpam-5887	333	23	)	)	PUNCT
ejpam-5887	333	24	;	;	PUNCT
ejpam-5887	333	25	0	0	NUM
ejpam-5887	333	26	≤	≤	NUM
ejpam-5887	333	27	r	r	NOUN
ejpam-5887	333	28	≤	≤	PUNCT
ejpam-5887	333	29	k−1	k−1	PROPN
ejpam-5887	333	30	as	as	SCONJ
ejpam-5887	333	31	follows	follow	VERB
ejpam-5887	333	32	:	:	PUNCT
ejpam-5887	333	33	f(e	f(e	NOUN
ejpam-5887	333	34	)	)	PUNCT
ejpam-5887	333	35	=	=	SYM
ejpam-5887	333	36	g(e	g(e	PROPN
ejpam-5887	333	37	)	)	PUNCT
ejpam-5887	333	38	;	;	PUNCT
ejpam-5887	334	1	e	e	X
ejpam-5887	334	2	∈	∈	PROPN
ejpam-5887	334	3	e(gi	e(gi	NOUN
ejpam-5887	334	4	)	)	PUNCT
ejpam-5887	334	5	,	,	PUNCT
ejpam-5887	334	6	1	1	NUM
ejpam-5887	334	7	≤	≤	NUM
ejpam-5887	334	8	i	i	PRON
ejpam-5887	334	9	≤	≤	PROPN
ejpam-5887	334	10	n	n	CCONJ
ejpam-5887	334	11	,	,	PUNCT
ejpam-5887	334	12	f(ei	f(ei	PROPN
ejpam-5887	334	13	)	)	PUNCT
ejpam-5887	334	14	=	=	PUNCT
ejpam-5887	334	15			NUM
ejpam-5887	334	16	0	0	NUM
ejpam-5887	334	17	;	;	PUNCT
ejpam-5887	334	18	1	1	NUM
ejpam-5887	334	19	≤	≤	NUM
ejpam-5887	334	20	i	i	PRON
ejpam-5887	334	21	≤	≤	PROPN
ejpam-5887	334	22	⌊n−1	⌊n−1	VERB
ejpam-5887	334	23	k	k	PROPN
ejpam-5887	334	24	⌋	⌋	PROPN
ejpam-5887	334	25	1	1	NUM
ejpam-5887	334	26	;	;	PUNCT
ejpam-5887	334	27	⌊n−1	⌊n−1	PROPN
ejpam-5887	334	28	k	k	X
ejpam-5887	334	29	⌋+	⌋+	ADJ
ejpam-5887	334	30	1	1	X
ejpam-5887	334	31	≤	≤	NUM
ejpam-5887	334	32	i	i	PRON
ejpam-5887	334	33	≤	≤	VERB
ejpam-5887	334	34	2⌊n−1	2⌊n−1	NUM
ejpam-5887	334	35	k	k	PROPN
ejpam-5887	334	36	⌋	⌋	PROPN
ejpam-5887	334	37	2	2	NUM
ejpam-5887	334	38	;	;	PUNCT
ejpam-5887	334	39	2⌊n−1	2⌊n−1	NUM
ejpam-5887	334	40	k	k	NOUN
ejpam-5887	334	41	⌋+	⌋+	PUNCT
ejpam-5887	334	42	1	1	NUM
ejpam-5887	334	43	≤	≤	NUM
ejpam-5887	334	44	i	i	PRON
ejpam-5887	334	45	≤	≤	PROPN
ejpam-5887	334	46	3⌊n−1	3⌊n−1	NUM
ejpam-5887	334	47	k	k	PROPN
ejpam-5887	334	48	⌋	⌋	NOUN
ejpam-5887	334	49	:	:	PUNCT
ejpam-5887	334	50	:	:	PUNCT
ejpam-5887	335	1	k	k	X
ejpam-5887	335	2	−	−	PROPN
ejpam-5887	335	3	1	1	NUM
ejpam-5887	335	4	;	;	PUNCT
ejpam-5887	335	5	(	(	PUNCT
ejpam-5887	335	6	k	k	X
ejpam-5887	335	7	−	−	PROPN
ejpam-5887	335	8	1)⌊n−1	1)⌊n−1	NUM
ejpam-5887	335	9	k	k	NOUN
ejpam-5887	335	10	⌋+	⌋+	ADJ
ejpam-5887	335	11	1	1	NUM
ejpam-5887	335	12	≤	≤	NUM
ejpam-5887	335	13	i	i	PRON
ejpam-5887	335	14	≤	≤	PROPN
ejpam-5887	335	15	k⌊n−1	k⌊n−1	VERB
ejpam-5887	335	16	k	k	PROPN
ejpam-5887	335	17	⌋	⌋	PROPN
ejpam-5887	335	18	,	,	PUNCT
ejpam-5887	335	19	f(ek⌊n−1	f(ek⌊n−1	PROPN
ejpam-5887	335	20	k	k	X
ejpam-5887	335	21	⌋+i	⌋+i	PROPN
ejpam-5887	335	22	)	)	PUNCT
ejpam-5887	336	1	=	=	SYM
ejpam-5887	336	2	j	j	PROPN
ejpam-5887	336	3	;	;	PUNCT
ejpam-5887	336	4	i	i	PROPN
ejpam-5887	336	5	≡	≡	PROPN
ejpam-5887	336	6	j	j	PROPN
ejpam-5887	336	7	(	(	PUNCT
ejpam-5887	336	8	mod	mod	PROPN
ejpam-5887	336	9	k	k	PROPN
ejpam-5887	336	10	)	)	PUNCT
ejpam-5887	336	11	,	,	PUNCT
ejpam-5887	336	12	1	1	NUM
ejpam-5887	336	13	≤	≤	NUM
ejpam-5887	336	14	i	i	PRON
ejpam-5887	336	15	≤	≤	ADJ
ejpam-5887	336	16	n−	n−	PROPN
ejpam-5887	336	17	1−	1−	PROPN
ejpam-5887	336	18	k⌊n−1	k⌊n−1	VERB
ejpam-5887	336	19	k	k	PROPN
ejpam-5887	336	20	⌋.	⌋.	ADV
ejpam-5887	336	21	from	from	ADP
ejpam-5887	336	22	this	this	DET
ejpam-5887	336	23	labeling	labeling	NOUN
ejpam-5887	336	24	we	we	PRON
ejpam-5887	336	25	obtain	obtain	VERB
ejpam-5887	336	26	,	,	PUNCT
ejpam-5887	336	27	ef	ef	PROPN
ejpam-5887	336	28	(	(	PUNCT
ejpam-5887	336	29	i	i	NOUN
ejpam-5887	336	30	)	)	PUNCT
ejpam-5887	336	31	=	=	PRON
ejpam-5887	336	32	{	{	PUNCT
ejpam-5887	336	33	neg(i	neg(i	PROPN
ejpam-5887	336	34	)	)	PUNCT
ejpam-5887	336	35	+	+	NUM
ejpam-5887	336	36	⌊n−1	⌊n−1	PROPN
ejpam-5887	336	37	k	k	PROPN
ejpam-5887	336	38	⌋	⌋	NOUN
ejpam-5887	336	39	;	;	PUNCT
ejpam-5887	336	40	i	i	PRON
ejpam-5887	336	41	≥	≥	VERB
ejpam-5887	336	42	r	r	NOUN
ejpam-5887	336	43	neg(i	neg(i	PROPN
ejpam-5887	336	44	)	)	PUNCT
ejpam-5887	336	45	+	+	NUM
ejpam-5887	336	46	⌊n−1	⌊n−1	PROPN
ejpam-5887	336	47	k	k	X
ejpam-5887	336	48	⌋+	⌋+	ADJ
ejpam-5887	336	49	1	1	NUM
ejpam-5887	336	50	;	;	PUNCT
ejpam-5887	336	51	i	i	PRON
ejpam-5887	336	52	<	<	X
ejpam-5887	336	53	r	r	NOUN
ejpam-5887	336	54	,	,	PUNCT
ejpam-5887	336	55	vf∗(i	vf∗(i	NOUN
ejpam-5887	336	56	)	)	PUNCT
ejpam-5887	336	57	=	=	SYM
ejpam-5887	336	58	vg∗(i	vg∗(i	ADJ
ejpam-5887	336	59	)	)	PUNCT
ejpam-5887	336	60	.	.	PUNCT
ejpam-5887	337	1	clearly	clearly	ADV
ejpam-5887	337	2	,	,	PUNCT
ejpam-5887	337	3	|ef	|ef	PROPN
ejpam-5887	337	4	(	(	PUNCT
ejpam-5887	337	5	i	i	NOUN
ejpam-5887	337	6	)	)	PUNCT
ejpam-5887	337	7	−	−	PROPN
ejpam-5887	337	8	ef	ef	PROPN
ejpam-5887	337	9	(	(	PUNCT
ejpam-5887	337	10	j)|	j)|	PROPN
ejpam-5887	337	11	≤	≤	NUM
ejpam-5887	337	12	1	1	NUM
ejpam-5887	337	13	and	and	CCONJ
ejpam-5887	337	14	|vf∗(i	|vf∗(i	NUM
ejpam-5887	337	15	)	)	PUNCT
ejpam-5887	337	16	−	−	NOUN
ejpam-5887	337	17	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	337	18	≤	≤	NOUN
ejpam-5887	337	19	1	1	NUM
ejpam-5887	337	20	for	for	ADP
ejpam-5887	337	21	i	i	PRON
ejpam-5887	337	22	,	,	PUNCT
ejpam-5887	337	23	j	j	PROPN
ejpam-5887	337	24	∈	∈	PROPN
ejpam-5887	337	25	{	{	PUNCT
ejpam-5887	337	26	0	0	NUM
ejpam-5887	337	27	,	,	PUNCT
ejpam-5887	337	28	1	1	NUM
ejpam-5887	337	29	,	,	PUNCT
ejpam-5887	337	30	2	2	NUM
ejpam-5887	337	31	..	..	PUNCT
ejpam-5887	337	32	,	,	PUNCT
ejpam-5887	337	33	k	k	PROPN
ejpam-5887	338	1	−	−	PROPN
ejpam-5887	338	2	1	1	NUM
ejpam-5887	338	3	}	}	PUNCT
ejpam-5887	338	4	.	.	PUNCT
ejpam-5887	339	1	hence	hence	ADV
ejpam-5887	339	2	,	,	PUNCT
ejpam-5887	339	3	p	p	X
ejpam-5887	339	4	(	(	PUNCT
ejpam-5887	339	5	n.gv	n.gv	NOUN
ejpam-5887	339	6	)	)	PUNCT
ejpam-5887	339	7	is	be	AUX
ejpam-5887	339	8	an	an	DET
ejpam-5887	339	9	edge	edge	NOUN
ejpam-5887	339	10	k	k	NOUN
ejpam-5887	339	11	-	-	PUNCT
ejpam-5887	339	12	product	product	NOUN
ejpam-5887	339	13	cordial	cordial	ADJ
ejpam-5887	339	14	graph	graph	NOUN
ejpam-5887	339	15	.	.	PUNCT
ejpam-5887	340	1	4.1	4.1	NUM
ejpam-5887	340	2	.	.	PUNCT
ejpam-5887	340	3	path	path	PROPN
ejpam-5887	340	4	union	union	PROPN
ejpam-5887	340	5	of	of	ADP
ejpam-5887	340	6	star	star	NOUN
ejpam-5887	340	7	in	in	ADP
ejpam-5887	340	8	this	this	DET
ejpam-5887	340	9	subsection	subsection	NOUN
ejpam-5887	340	10	,	,	PUNCT
ejpam-5887	340	11	we	we	PRON
ejpam-5887	340	12	prove	prove	VERB
ejpam-5887	340	13	that	that	SCONJ
ejpam-5887	340	14	the	the	DET
ejpam-5887	340	15	path	path	NOUN
ejpam-5887	340	16	union	union	NOUN
ejpam-5887	340	17	of	of	ADP
ejpam-5887	340	18	a	a	DET
ejpam-5887	340	19	star	star	NOUN
ejpam-5887	340	20	graph	graph	NOUN
ejpam-5887	340	21	p	p	X
ejpam-5887	340	22	(	(	PUNCT
ejpam-5887	340	23	n.kv	n.kv	PROPN
ejpam-5887	340	24	1,m	1,m	NOUN
ejpam-5887	340	25	)	)	PUNCT
ejpam-5887	340	26	,	,	PUNCT
ejpam-5887	340	27	where	where	SCONJ
ejpam-5887	340	28	v	v	NOUN
ejpam-5887	340	29	is	be	AUX
ejpam-5887	340	30	a	a	DET
ejpam-5887	340	31	root	root	NOUN
ejpam-5887	340	32	vertex	vertex	NOUN
ejpam-5887	340	33	of	of	ADP
ejpam-5887	340	34	k1,m	k1,m	PROPN
ejpam-5887	340	35	admits	admit	VERB
ejpam-5887	340	36	an	an	DET
ejpam-5887	340	37	edge	edge	NOUN
ejpam-5887	340	38	k	k	NOUN
ejpam-5887	340	39	-	-	PUNCT
ejpam-5887	340	40	product	product	NOUN
ejpam-5887	340	41	cordial	cordial	ADJ
ejpam-5887	340	42	labeling	labeling	NOUN
ejpam-5887	340	43	for	for	ADP
ejpam-5887	340	44	n	n	X
ejpam-5887	340	45	≡	≡	PROPN
ejpam-5887	340	46	0	0	NUM
ejpam-5887	340	47	,	,	PUNCT
ejpam-5887	340	48	1	1	NUM
ejpam-5887	340	49	(	(	PUNCT
ejpam-5887	340	50	mod	mod	NOUN
ejpam-5887	340	51	k	k	PROPN
ejpam-5887	340	52	)	)	PUNCT
ejpam-5887	340	53	.	.	PUNCT
ejpam-5887	341	1	also	also	ADV
ejpam-5887	341	2	,	,	PUNCT
ejpam-5887	341	3	we	we	PRON
ejpam-5887	341	4	show	show	VERB
ejpam-5887	341	5	that	that	SCONJ
ejpam-5887	341	6	the	the	DET
ejpam-5887	341	7	graph	graph	NOUN
ejpam-5887	341	8	p	p	X
ejpam-5887	341	9	(	(	PUNCT
ejpam-5887	341	10	n.kv	n.kv	PROPN
ejpam-5887	341	11	1,m	1,m	NOUN
ejpam-5887	341	12	)	)	PUNCT
ejpam-5887	341	13	admits	admit	VERB
ejpam-5887	341	14	an	an	DET
ejpam-5887	341	15	edge	edge	NOUN
ejpam-5887	341	16	k	k	NOUN
ejpam-5887	341	17	-	-	PUNCT
ejpam-5887	341	18	product	product	NOUN
ejpam-5887	341	19	cordial	cordial	ADJ
ejpam-5887	341	20	labeling	labeling	NOUN
ejpam-5887	341	21	for	for	ADP
ejpam-5887	341	22	n	n	PRON
ejpam-5887	341	23	≡	≡	PROPN
ejpam-5887	341	24	k	k	PROPN
ejpam-5887	342	1	−	−	PROPN
ejpam-5887	342	2	1	1	NUM
ejpam-5887	342	3	(	(	PUNCT
ejpam-5887	342	4	mod	mod	PROPN
ejpam-5887	342	5	k	k	PROPN
ejpam-5887	342	6	)	)	PUNCT
ejpam-5887	342	7	if	if	SCONJ
ejpam-5887	342	8	m	m	VERB
ejpam-5887	342	9	≡	≡	PROPN
ejpam-5887	342	10	0	0	NUM
ejpam-5887	342	11	,	,	PUNCT
ejpam-5887	342	12	k	k	PROPN
ejpam-5887	343	1	−	−	PROPN
ejpam-5887	343	2	1	1	NUM
ejpam-5887	343	3	(	(	PUNCT
ejpam-5887	343	4	mod	mod	PROPN
ejpam-5887	343	5	k	k	PROPN
ejpam-5887	343	6	)	)	PUNCT
ejpam-5887	343	7	.	.	PUNCT
ejpam-5887	344	1	n.	n.	PROPN
ejpam-5887	344	2	m.	m.	PROPN
ejpam-5887	344	3	noureldeen	noureldeen	INTJ
ejpam-5887	344	4	et	et	PROPN
ejpam-5887	344	5	al	al	PROPN
ejpam-5887	344	6	.	.	PUNCT
ejpam-5887	344	7	/	/	SYM
ejpam-5887	344	8	eur	eur	PROPN
ejpam-5887	344	9	.	.	PUNCT
ejpam-5887	345	1	j.	j.	PROPN
ejpam-5887	345	2	pure	pure	PROPN
ejpam-5887	345	3	appl	appl	PROPN
ejpam-5887	345	4	.	.	PROPN
ejpam-5887	345	5	math	math	PROPN
ejpam-5887	345	6	,	,	PUNCT
ejpam-5887	345	7	18	18	NUM
ejpam-5887	345	8	(	(	PUNCT
ejpam-5887	345	9	2	2	NUM
ejpam-5887	345	10	)	)	PUNCT
ejpam-5887	345	11	(	(	PUNCT
ejpam-5887	345	12	2025	2025	NUM
ejpam-5887	345	13	)	)	PUNCT
ejpam-5887	345	14	,	,	PUNCT
ejpam-5887	345	15	5887	5887	NUM
ejpam-5887	345	16	11	11	NUM
ejpam-5887	345	17	of	of	ADP
ejpam-5887	345	18	21	21	NUM
ejpam-5887	345	19	theorem	theorem	VERB
ejpam-5887	345	20	15	15	NUM
ejpam-5887	345	21	.	.	PUNCT
ejpam-5887	346	1	the	the	DET
ejpam-5887	346	2	path	path	PROPN
ejpam-5887	346	3	union	union	PROPN
ejpam-5887	346	4	of	of	ADP
ejpam-5887	346	5	star	star	PROPN
ejpam-5887	346	6	graph	graph	NOUN
ejpam-5887	346	7	p	p	X
ejpam-5887	346	8	(	(	PUNCT
ejpam-5887	346	9	n.kv	n.kv	PROPN
ejpam-5887	346	10	1,m	1,m	NOUN
ejpam-5887	346	11	)	)	PUNCT
ejpam-5887	346	12	,	,	PUNCT
ejpam-5887	346	13	where	where	SCONJ
ejpam-5887	346	14	v	v	NOUN
ejpam-5887	346	15	is	be	AUX
ejpam-5887	346	16	a	a	DET
ejpam-5887	346	17	root	root	NOUN
ejpam-5887	346	18	vertex	vertex	NOUN
ejpam-5887	346	19	of	of	ADP
ejpam-5887	346	20	k1,m	k1,m	PROPN
ejpam-5887	346	21	admits	admit	VERB
ejpam-5887	346	22	an	an	DET
ejpam-5887	346	23	edge	edge	NOUN
ejpam-5887	346	24	k	k	NOUN
ejpam-5887	346	25	-	-	PUNCT
ejpam-5887	346	26	product	product	NOUN
ejpam-5887	346	27	cordial	cordial	ADJ
ejpam-5887	346	28	labeling	labeling	NOUN
ejpam-5887	346	29	if	if	SCONJ
ejpam-5887	346	30	n	n	PRON
ejpam-5887	346	31	≡	≡	PROPN
ejpam-5887	346	32	0	0	NUM
ejpam-5887	346	33	,	,	PUNCT
ejpam-5887	346	34	1	1	NUM
ejpam-5887	346	35	(	(	PUNCT
ejpam-5887	346	36	mod	mod	NOUN
ejpam-5887	346	37	k	k	PROPN
ejpam-5887	346	38	)	)	PUNCT
ejpam-5887	346	39	.	.	PUNCT
ejpam-5887	347	1	proof	proof	NOUN
ejpam-5887	347	2	.	.	PUNCT
ejpam-5887	348	1	let	let	VERB
ejpam-5887	348	2	the	the	DET
ejpam-5887	348	3	vertex	vertex	NOUN
ejpam-5887	348	4	and	and	CCONJ
ejpam-5887	348	5	edge	edge	NOUN
ejpam-5887	348	6	set	set	NOUN
ejpam-5887	348	7	of	of	ADP
ejpam-5887	348	8	p	p	X
ejpam-5887	348	9	(	(	PUNCT
ejpam-5887	348	10	n.kv	n.kv	PROPN
ejpam-5887	348	11	1,m	1,m	NOUN
ejpam-5887	348	12	)	)	PUNCT
ejpam-5887	348	13	be	be	AUX
ejpam-5887	348	14	v	v	PRON
ejpam-5887	348	15	(	(	PUNCT
ejpam-5887	348	16	p	p	X
ejpam-5887	348	17	(	(	PUNCT
ejpam-5887	348	18	n.kv	n.kv	PROPN
ejpam-5887	348	19	1,m	1,m	NOUN
ejpam-5887	348	20	)	)	PUNCT
ejpam-5887	348	21	)	)	PUNCT
ejpam-5887	349	1	=	=	PRON
ejpam-5887	349	2	{	{	PUNCT
ejpam-5887	349	3	vi	vi	PROPN
ejpam-5887	349	4	,	,	PUNCT
ejpam-5887	349	5	vji	vji	NOUN
ejpam-5887	349	6	:	:	PUNCT
ejpam-5887	349	7	1	1	NUM
ejpam-5887	349	8	≤	≤	NUM
ejpam-5887	349	9	i	i	PRON
ejpam-5887	349	10	≤	≤	PROPN
ejpam-5887	349	11	n	n	CCONJ
ejpam-5887	349	12	,	,	PUNCT
ejpam-5887	349	13	1	1	NUM
ejpam-5887	349	14	≤	≤	NUM
ejpam-5887	350	1	j	j	PROPN
ejpam-5887	350	2	≤	≤	PROPN
ejpam-5887	350	3	m	m	PROPN
ejpam-5887	350	4	}	}	PUNCT
ejpam-5887	350	5	and	and	CCONJ
ejpam-5887	350	6	e(p	e(p	PROPN
ejpam-5887	350	7	(	(	PUNCT
ejpam-5887	350	8	n.kv	n.kv	PROPN
ejpam-5887	350	9	1,m	1,m	NOUN
ejpam-5887	350	10	)	)	PUNCT
ejpam-5887	350	11	)	)	PUNCT
ejpam-5887	351	1	=	=	PRON
ejpam-5887	351	2	{	{	PUNCT
ejpam-5887	351	3	vivi+1	vivi+1	PROPN
ejpam-5887	351	4	,	,	PUNCT
ejpam-5887	351	5	viv	viv	PROPN
ejpam-5887	351	6	j	j	PROPN
ejpam-5887	352	1	i	i	PROPN
ejpam-5887	352	2	,	,	PUNCT
ejpam-5887	352	3	vnv	vnv	NOUN
ejpam-5887	352	4	j	j	PROPN
ejpam-5887	352	5	n	n	CCONJ
ejpam-5887	352	6	:	:	PUNCT
ejpam-5887	352	7	1	1	NUM
ejpam-5887	352	8	≤	≤	NUM
ejpam-5887	352	9	i	i	PRON
ejpam-5887	352	10	≤	≤	ADJ
ejpam-5887	352	11	n	n	CCONJ
ejpam-5887	352	12	−	−	PROPN
ejpam-5887	352	13	1	1	NUM
ejpam-5887	352	14	,	,	PUNCT
ejpam-5887	352	15	1	1	NUM
ejpam-5887	352	16	≤	≤	NUM
ejpam-5887	352	17	j	j	PROPN
ejpam-5887	352	18	≤	≤	PROPN
ejpam-5887	352	19	m	m	VERB
ejpam-5887	352	20	}	}	PUNCT
ejpam-5887	352	21	respectively	respectively	ADV
ejpam-5887	352	22	.	.	PUNCT
ejpam-5887	353	1	if	if	SCONJ
ejpam-5887	353	2	m	m	VERB
ejpam-5887	353	3	≡	≡	PROPN
ejpam-5887	353	4	0	0	PUNCT
ejpam-5887	354	1	(	(	PUNCT
ejpam-5887	354	2	mod	mod	PROPN
ejpam-5887	354	3	k	k	PROPN
ejpam-5887	354	4	)	)	PUNCT
ejpam-5887	354	5	,	,	PUNCT
ejpam-5887	354	6	by	by	ADP
ejpam-5887	354	7	theorems	theorem	NOUN
ejpam-5887	354	8	1	1	NUM
ejpam-5887	354	9	and	and	CCONJ
ejpam-5887	354	10	14	14	NUM
ejpam-5887	354	11	,	,	PUNCT
ejpam-5887	354	12	p	p	X
ejpam-5887	354	13	(	(	PUNCT
ejpam-5887	354	14	n.kv	n.kv	PROPN
ejpam-5887	354	15	1,m	1,m	NOUN
ejpam-5887	354	16	)	)	PUNCT
ejpam-5887	354	17	is	be	AUX
ejpam-5887	354	18	an	an	DET
ejpam-5887	354	19	edge	edge	NOUN
ejpam-5887	354	20	k	k	NOUN
ejpam-5887	354	21	-	-	PUNCT
ejpam-5887	354	22	product	product	NOUN
ejpam-5887	354	23	cordial	cordial	ADJ
ejpam-5887	354	24	graph	graph	NOUN
ejpam-5887	354	25	.	.	PUNCT
ejpam-5887	355	1	define	define	VERB
ejpam-5887	355	2	f	f	PROPN
ejpam-5887	355	3	:	:	PUNCT
ejpam-5887	355	4	e(p	e(p	PROPN
ejpam-5887	355	5	(	(	PUNCT
ejpam-5887	355	6	n.kv	n.kv	PROPN
ejpam-5887	355	7	1,m	1,m	NOUN
ejpam-5887	355	8	)	)	PUNCT
ejpam-5887	355	9	)	)	PUNCT
ejpam-5887	356	1	→	→	PUNCT
ejpam-5887	356	2	{	{	PUNCT
ejpam-5887	356	3	0	0	NUM
ejpam-5887	356	4	,	,	PUNCT
ejpam-5887	356	5	1	1	NUM
ejpam-5887	356	6	,	,	PUNCT
ejpam-5887	356	7	2	2	NUM
ejpam-5887	356	8	,	,	PUNCT
ejpam-5887	356	9	...	...	PUNCT
ejpam-5887	356	10	,	,	PUNCT
ejpam-5887	356	11	k−1	k−1	PROPN
ejpam-5887	356	12	}	}	PUNCT
ejpam-5887	356	13	for	for	ADP
ejpam-5887	356	14	n	n	X
ejpam-5887	356	15	≡	≡	PROPN
ejpam-5887	356	16	0	0	NUM
ejpam-5887	356	17	,	,	PUNCT
ejpam-5887	356	18	1	1	NUM
ejpam-5887	356	19	(	(	PUNCT
ejpam-5887	356	20	mod	mod	PROPN
ejpam-5887	356	21	k	k	PROPN
ejpam-5887	356	22	)	)	PUNCT
ejpam-5887	356	23	andm	andm	PROPN
ejpam-5887	356	24	≡	≡	PROPN
ejpam-5887	356	25	r	r	PROPN
ejpam-5887	356	26	(	(	PUNCT
ejpam-5887	356	27	mod	mod	PROPN
ejpam-5887	356	28	k	k	PROPN
ejpam-5887	356	29	)	)	PUNCT
ejpam-5887	356	30	;	;	PUNCT
ejpam-5887	356	31	1	1	NUM
ejpam-5887	356	32	≤	≤	NUM
ejpam-5887	356	33	r	r	NOUN
ejpam-5887	356	34	≤	≤	PUNCT
ejpam-5887	356	35	k	k	NOUN
ejpam-5887	356	36	−	−	NOUN
ejpam-5887	356	37	1	1	NUM
ejpam-5887	356	38	as	as	SCONJ
ejpam-5887	356	39	follows	follow	VERB
ejpam-5887	356	40	:	:	PUNCT
ejpam-5887	356	41	f(vivi+1	f(vivi+1	X
ejpam-5887	356	42	)	)	PUNCT
ejpam-5887	356	43	=	=	SYM
ejpam-5887	356	44	0	0	NUM
ejpam-5887	356	45	;	;	PUNCT
ejpam-5887	356	46	1	1	NUM
ejpam-5887	356	47	≤	≤	NUM
ejpam-5887	357	1	i	i	PRON
ejpam-5887	357	2	≤	≤	ADJ
ejpam-5887	357	3	n−	n−	NOUN
ejpam-5887	357	4	1	1	NUM
ejpam-5887	357	5	,	,	PUNCT
ejpam-5887	357	6	we	we	PRON
ejpam-5887	357	7	consider	consider	VERB
ejpam-5887	357	8	the	the	DET
ejpam-5887	357	9	following	follow	VERB
ejpam-5887	357	10	two	two	NUM
ejpam-5887	357	11	cases	case	NOUN
ejpam-5887	357	12	.	.	PUNCT
ejpam-5887	358	1	case(i	case(i	NOUN
ejpam-5887	358	2	):	):	PUNCT
ejpam-5887	358	3	if	if	SCONJ
ejpam-5887	358	4	n	n	PRON
ejpam-5887	358	5	≡	≡	PROPN
ejpam-5887	358	6	0	0	PUNCT
ejpam-5887	359	1	(	(	PUNCT
ejpam-5887	359	2	mod	mod	PROPN
ejpam-5887	359	3	k	k	PROPN
ejpam-5887	359	4	)	)	PUNCT
ejpam-5887	359	5	,	,	PUNCT
ejpam-5887	359	6	then	then	ADV
ejpam-5887	359	7	f(viv	f(viv	PROPN
ejpam-5887	360	1	j	j	PROPN
ejpam-5887	361	1	i	i	NOUN
ejpam-5887	361	2	)	)	PUNCT
ejpam-5887	362	1	=	=	PUNCT
ejpam-5887	362	2			NUM
ejpam-5887	362	3	0	0	NUM
ejpam-5887	362	4	;	;	PUNCT
ejpam-5887	362	5	1	1	NUM
ejpam-5887	362	6	≤	≤	NUM
ejpam-5887	362	7	i	i	PRON
ejpam-5887	362	8	≤	≤	NOUN
ejpam-5887	363	1	n	n	CCONJ
ejpam-5887	363	2	k	k	NOUN
ejpam-5887	363	3	,	,	PUNCT
ejpam-5887	363	4	1	1	NUM
ejpam-5887	363	5	≤	≤	NUM
ejpam-5887	363	6	j	j	PROPN
ejpam-5887	363	7	≤	≤	PROPN
ejpam-5887	363	8	k⌊mk	k⌊mk	NOUN
ejpam-5887	363	9	⌋	⌋	NOUN
ejpam-5887	363	10	−	−	PROPN
ejpam-5887	364	1	(	(	PUNCT
ejpam-5887	364	2	k	k	NOUN
ejpam-5887	364	3	−	−	PROPN
ejpam-5887	364	4	1	1	NUM
ejpam-5887	364	5	)	)	PUNCT
ejpam-5887	365	1	+	+	CCONJ
ejpam-5887	365	2	r	r	NOUN
ejpam-5887	365	3	1	1	NUM
ejpam-5887	365	4	;	;	PUNCT
ejpam-5887	365	5	1	1	NUM
ejpam-5887	365	6	≤	≤	NUM
ejpam-5887	365	7	i	i	PRON
ejpam-5887	365	8	≤	≤	NOUN
ejpam-5887	366	1	n	n	PRON
ejpam-5887	366	2	k	k	NOUN
ejpam-5887	366	3	,	,	PUNCT
ejpam-5887	366	4	j	j	PROPN
ejpam-5887	366	5	=	=	SYM
ejpam-5887	366	6	k⌊mk	k⌊mk	X
ejpam-5887	366	7	⌋	⌋	NOUN
ejpam-5887	366	8	−	−	PROPN
ejpam-5887	367	1	(	(	PUNCT
ejpam-5887	367	2	k	k	NOUN
ejpam-5887	367	3	−	−	PROPN
ejpam-5887	367	4	1	1	NUM
ejpam-5887	367	5	)	)	PUNCT
ejpam-5887	367	6	+	+	CCONJ
ejpam-5887	367	7	r	r	NOUN
ejpam-5887	367	8	+	+	ADJ
ejpam-5887	367	9	1	1	NUM
ejpam-5887	367	10	;	;	PUNCT
ejpam-5887	367	11	n	n	PRON
ejpam-5887	367	12	k	k	NOUN
ejpam-5887	368	1	+	+	CCONJ
ejpam-5887	368	2	1	1	X
ejpam-5887	368	3	≤	≤	NUM
ejpam-5887	368	4	i	i	PRON
ejpam-5887	368	5	≤	≤	NOUN
ejpam-5887	368	6	2n	2n	NUM
ejpam-5887	368	7	k	k	PROPN
ejpam-5887	368	8	2	2	NUM
ejpam-5887	368	9	;	;	PUNCT
ejpam-5887	368	10	1	1	NUM
ejpam-5887	368	11	≤	≤	NUM
ejpam-5887	368	12	i	i	PRON
ejpam-5887	368	13	≤	≤	NOUN
ejpam-5887	369	1	n	n	PRON
ejpam-5887	369	2	k	k	NOUN
ejpam-5887	369	3	,	,	PUNCT
ejpam-5887	369	4	j	j	PROPN
ejpam-5887	369	5	=	=	SYM
ejpam-5887	369	6	k⌊mk	k⌊mk	X
ejpam-5887	369	7	⌋	⌋	NOUN
ejpam-5887	369	8	−	−	PROPN
ejpam-5887	370	1	(	(	PUNCT
ejpam-5887	370	2	k	k	NOUN
ejpam-5887	370	3	−	−	PROPN
ejpam-5887	370	4	1	1	NUM
ejpam-5887	370	5	)	)	PUNCT
ejpam-5887	370	6	+	+	CCONJ
ejpam-5887	370	7	r	r	NOUN
ejpam-5887	370	8	+	+	ADJ
ejpam-5887	370	9	2	2	NUM
ejpam-5887	370	10	;	;	PUNCT
ejpam-5887	370	11	2n	2n	NUM
ejpam-5887	370	12	k	k	NOUN
ejpam-5887	371	1	+	+	CCONJ
ejpam-5887	371	2	1	1	X
ejpam-5887	371	3	≤	≤	NUM
ejpam-5887	371	4	i	i	PRON
ejpam-5887	371	5	≤	≤	ADJ
ejpam-5887	371	6	3n	3n	NUM
ejpam-5887	371	7	k	k	NOUN
ejpam-5887	371	8	:	:	PUNCT
ejpam-5887	371	9	:	:	PUNCT
ejpam-5887	372	1	k	k	X
ejpam-5887	372	2	−	−	PROPN
ejpam-5887	372	3	1	1	NUM
ejpam-5887	372	4	;	;	PUNCT
ejpam-5887	372	5	1	1	NUM
ejpam-5887	372	6	≤	≤	NUM
ejpam-5887	372	7	i	i	PRON
ejpam-5887	372	8	≤	≤	NOUN
ejpam-5887	372	9	n	n	PRON
ejpam-5887	372	10	k	k	NOUN
ejpam-5887	372	11	,	,	PUNCT
ejpam-5887	372	12	j	j	PROPN
ejpam-5887	372	13	=	=	PUNCT
ejpam-5887	372	14	k⌊mk	k⌊mk	X
ejpam-5887	372	15	⌋+	⌋+	VERB
ejpam-5887	372	16	r	r	NOUN
ejpam-5887	372	17	;	;	PUNCT
ejpam-5887	372	18	(	(	PUNCT
ejpam-5887	372	19	k−1)n	k−1)n	PROPN
ejpam-5887	372	20	k	k	PROPN
ejpam-5887	372	21	+	+	CCONJ
ejpam-5887	372	22	1	1	X
ejpam-5887	372	23	≤	≤	NUM
ejpam-5887	372	24	i	i	PRON
ejpam-5887	372	25	≤	≤	ADJ
ejpam-5887	372	26	n.	n.	NOUN
ejpam-5887	372	27	from	from	ADP
ejpam-5887	372	28	this	this	DET
ejpam-5887	372	29	labeling	labeling	NOUN
ejpam-5887	372	30	we	we	PRON
ejpam-5887	372	31	get	get	VERB
ejpam-5887	372	32	,	,	PUNCT
ejpam-5887	372	33	ef	ef	PROPN
ejpam-5887	372	34	(	(	PUNCT
ejpam-5887	372	35	i	i	NOUN
ejpam-5887	372	36	)	)	PUNCT
ejpam-5887	372	37	=	=	PRON
ejpam-5887	372	38	{	{	PUNCT
ejpam-5887	372	39	n⌊mk	n⌊mk	NOUN
ejpam-5887	372	40	⌋+	⌋+	X
ejpam-5887	373	1	n	n	CCONJ
ejpam-5887	373	2	k	k	X
ejpam-5887	373	3	(	(	PUNCT
ejpam-5887	373	4	1	1	NUM
ejpam-5887	373	5	+	+	CCONJ
ejpam-5887	373	6	r)−	r)−	PROPN
ejpam-5887	373	7	1	1	NUM
ejpam-5887	373	8	;	;	PUNCT
ejpam-5887	373	9	i	i	PRON
ejpam-5887	373	10	=	=	SYM
ejpam-5887	373	11	0	0	NUM
ejpam-5887	373	12	n⌊mk	n⌊mk	NOUN
ejpam-5887	373	13	⌋+	⌋+	PUNCT
ejpam-5887	374	1	n	n	CCONJ
ejpam-5887	374	2	k	k	X
ejpam-5887	374	3	(	(	PUNCT
ejpam-5887	374	4	1	1	NUM
ejpam-5887	374	5	+	+	CCONJ
ejpam-5887	374	6	r	r	NOUN
ejpam-5887	374	7	)	)	PUNCT
ejpam-5887	374	8	;	;	PUNCT
ejpam-5887	374	9	1	1	NUM
ejpam-5887	374	10	≤	≤	NUM
ejpam-5887	374	11	i	i	X
ejpam-5887	374	12	≤	≤	NOUN
ejpam-5887	375	1	k	k	PRON
ejpam-5887	375	2	−	−	PROPN
ejpam-5887	375	3	1	1	NUM
ejpam-5887	375	4	,	,	PUNCT
ejpam-5887	375	5	vf∗(i	vf∗(i	ADJ
ejpam-5887	375	6	)	)	PUNCT
ejpam-5887	375	7	=	=	PUNCT
ejpam-5887	376	1	n⌊mk	n⌊mk	NOUN
ejpam-5887	376	2	⌋+	⌋+	PUNCT
ejpam-5887	377	1	n	n	CCONJ
ejpam-5887	377	2	k	k	X
ejpam-5887	377	3	(	(	PUNCT
ejpam-5887	377	4	1	1	NUM
ejpam-5887	377	5	+	+	CCONJ
ejpam-5887	377	6	r	r	NOUN
ejpam-5887	377	7	)	)	PUNCT
ejpam-5887	377	8	;	;	PUNCT
ejpam-5887	377	9	0	0	NUM
ejpam-5887	377	10	≤	≤	NUM
ejpam-5887	378	1	i	i	PRON
ejpam-5887	378	2	≤	≤	NOUN
ejpam-5887	379	1	k	k	PRON
ejpam-5887	380	1	−	−	NOUN
ejpam-5887	380	2	1	1	X
ejpam-5887	380	3	.	.	PUNCT
ejpam-5887	380	4	case	case	NOUN
ejpam-5887	380	5	(	(	PUNCT
ejpam-5887	380	6	ii	ii	NOUN
ejpam-5887	380	7	):	):	PUNCT
ejpam-5887	380	8	if	if	SCONJ
ejpam-5887	380	9	n	n	PRON
ejpam-5887	380	10	≡	≡	PROPN
ejpam-5887	380	11	1	1	NUM
ejpam-5887	380	12	(	(	PUNCT
ejpam-5887	380	13	mod	mod	PROPN
ejpam-5887	380	14	k	k	PROPN
ejpam-5887	380	15	)	)	PUNCT
ejpam-5887	380	16	,	,	PUNCT
ejpam-5887	380	17	then	then	ADV
ejpam-5887	380	18	f(viv	f(viv	PROPN
ejpam-5887	380	19	j	j	PROPN
ejpam-5887	380	20	i	i	PROPN
ejpam-5887	380	21	)	)	PUNCT
ejpam-5887	380	22	;	;	PUNCT
ejpam-5887	380	23	1	1	NUM
ejpam-5887	380	24	≤	≤	NUM
ejpam-5887	380	25	i	i	PRON
ejpam-5887	380	26	≤	≤	ADJ
ejpam-5887	380	27	n−	n−	NOUN
ejpam-5887	380	28	1	1	NUM
ejpam-5887	380	29	,	,	PUNCT
ejpam-5887	380	30	1	1	NUM
ejpam-5887	380	31	≤	≤	NUM
ejpam-5887	380	32	j	j	PROPN
ejpam-5887	380	33	≤	≤	NUM
ejpam-5887	380	34	m	m	VERB
ejpam-5887	380	35	as	as	ADP
ejpam-5887	380	36	in	in	ADP
ejpam-5887	380	37	case	case	NOUN
ejpam-5887	380	38	(	(	PUNCT
ejpam-5887	380	39	i	i	NOUN
ejpam-5887	380	40	)	)	PUNCT
ejpam-5887	380	41	,	,	PUNCT
ejpam-5887	380	42	f(vnv	f(vnv	PROPN
ejpam-5887	380	43	j	j	PROPN
ejpam-5887	380	44	n	n	CCONJ
ejpam-5887	380	45	)	)	PUNCT
ejpam-5887	380	46	=	=	PUNCT
ejpam-5887	381	1			NUM
ejpam-5887	381	2	0	0	NUM
ejpam-5887	381	3	;	;	PUNCT
ejpam-5887	381	4	1	1	NUM
ejpam-5887	381	5	≤	≤	NUM
ejpam-5887	381	6	j	j	PROPN
ejpam-5887	381	7	≤	≤	PROPN
ejpam-5887	381	8	⌊mk	⌊mk	NOUN
ejpam-5887	381	9	⌋	⌋	NOUN
ejpam-5887	381	10	1	1	NUM
ejpam-5887	381	11	;	;	PUNCT
ejpam-5887	381	12	⌊mk	⌊mk	X
ejpam-5887	381	13	⌋+	⌋+	PUNCT
ejpam-5887	381	14	1	1	NUM
ejpam-5887	381	15	≤	≤	NUM
ejpam-5887	381	16	j	j	PROPN
ejpam-5887	381	17	≤	≤	PROPN
ejpam-5887	381	18	2⌊mk	2⌊mk	NUM
ejpam-5887	381	19	⌋	⌋	NOUN
ejpam-5887	381	20	2	2	NUM
ejpam-5887	381	21	;	;	PUNCT
ejpam-5887	381	22	2⌊mk	2⌊mk	NUM
ejpam-5887	381	23	⌋+	⌋+	NUM
ejpam-5887	381	24	1	1	NUM
ejpam-5887	381	25	≤	≤	NUM
ejpam-5887	382	1	j	j	PROPN
ejpam-5887	382	2	≤	≤	PROPN
ejpam-5887	382	3	3⌊mk	3⌊mk	NUM
ejpam-5887	382	4	⌋	⌋	NOUN
ejpam-5887	382	5	:	:	PUNCT
ejpam-5887	382	6	:	:	PUNCT
ejpam-5887	383	1	k	k	X
ejpam-5887	383	2	−	−	PROPN
ejpam-5887	383	3	1	1	NUM
ejpam-5887	383	4	;	;	PUNCT
ejpam-5887	383	5	(	(	PUNCT
ejpam-5887	383	6	k	k	X
ejpam-5887	383	7	−	−	PROPN
ejpam-5887	383	8	1)⌊mk	1)⌊mk	NUM
ejpam-5887	383	9	⌋+	⌋+	NUM
ejpam-5887	383	10	1	1	NUM
ejpam-5887	383	11	≤	≤	NUM
ejpam-5887	383	12	j	j	PROPN
ejpam-5887	383	13	≤	≤	PROPN
ejpam-5887	383	14	k⌊mk	k⌊mk	NOUN
ejpam-5887	383	15	⌋	⌋	NOUN
ejpam-5887	383	16	,	,	PUNCT
ejpam-5887	383	17	f(vnv	f(vnv	PROPN
ejpam-5887	383	18	k⌊m	k⌊m	PROPN
ejpam-5887	383	19	k	k	PROPN
ejpam-5887	383	20	⌋+j	⌋+j	PROPN
ejpam-5887	383	21	n	n	CCONJ
ejpam-5887	383	22	)	)	PUNCT
ejpam-5887	383	23	=	=	SYM
ejpam-5887	383	24	j	j	PROPN
ejpam-5887	383	25	;	;	PUNCT
ejpam-5887	383	26	1	1	NUM
ejpam-5887	383	27	≤	≤	NUM
ejpam-5887	383	28	j	j	PROPN
ejpam-5887	383	29	≤	≤	PROPN
ejpam-5887	383	30	r.	r.	PROPN
ejpam-5887	383	31	from	from	ADP
ejpam-5887	383	32	this	this	DET
ejpam-5887	383	33	labeling	labeling	NOUN
ejpam-5887	383	34	we	we	PRON
ejpam-5887	383	35	have	have	VERB
ejpam-5887	383	36	,	,	PUNCT
ejpam-5887	383	37	ef	ef	PROPN
ejpam-5887	383	38	(	(	PUNCT
ejpam-5887	383	39	i	i	NOUN
ejpam-5887	383	40	)	)	PUNCT
ejpam-5887	383	41	=	=	PRON
ejpam-5887	383	42	{	{	PUNCT
ejpam-5887	383	43	⌊nk	⌊nk	NOUN
ejpam-5887	383	44	⌋+	⌋+	X
ejpam-5887	383	45	n⌊mk	n⌊mk	PROPN
ejpam-5887	383	46	⌋+	⌋+	X
ejpam-5887	383	47	r⌊nk	r⌊nk	VERB
ejpam-5887	383	48	⌋	⌋	NOUN
ejpam-5887	383	49	;	;	PUNCT
ejpam-5887	383	50	i	i	NOUN
ejpam-5887	383	51	=	=	NOUN
ejpam-5887	383	52	0	0	NUM
ejpam-5887	383	53	;	;	PUNCT
ejpam-5887	383	54	i	i	PRON
ejpam-5887	383	55	>	>	PUNCT
ejpam-5887	383	56	r	r	NOUN
ejpam-5887	383	57	⌊nk	⌊nk	NOUN
ejpam-5887	383	58	⌋+	⌋+	X
ejpam-5887	383	59	n⌊mk	n⌊mk	PROPN
ejpam-5887	383	60	⌋+	⌋+	X
ejpam-5887	383	61	r⌊nk	r⌊nk	VERB
ejpam-5887	383	62	⌋+	⌋+	X
ejpam-5887	383	63	1	1	NUM
ejpam-5887	383	64	;	;	PUNCT
ejpam-5887	383	65	1	1	NUM
ejpam-5887	383	66	≤	≤	NUM
ejpam-5887	383	67	i	i	X
ejpam-5887	383	68	≤	≤	ADJ
ejpam-5887	383	69	r	r	NOUN
ejpam-5887	383	70	,	,	PUNCT
ejpam-5887	383	71	vf∗(i	vf∗(i	NOUN
ejpam-5887	383	72	)	)	PUNCT
ejpam-5887	383	73	=	=	SYM
ejpam-5887	383	74	{	{	PUNCT
ejpam-5887	383	75	⌊nk	⌊nk	NOUN
ejpam-5887	383	76	⌋+	⌋+	X
ejpam-5887	383	77	n⌊mk	n⌊mk	PROPN
ejpam-5887	383	78	⌋+	⌋+	X
ejpam-5887	383	79	r⌊nk	r⌊nk	VERB
ejpam-5887	383	80	⌋	⌋	NOUN
ejpam-5887	383	81	;	;	PUNCT
ejpam-5887	383	82	i	i	PRON
ejpam-5887	383	83	>	>	PUNCT
ejpam-5887	383	84	r	r	NOUN
ejpam-5887	383	85	⌊nk	⌊nk	NOUN
ejpam-5887	383	86	⌋+	⌋+	X
ejpam-5887	383	87	n⌊mk	n⌊mk	PROPN
ejpam-5887	383	88	⌋+	⌋+	X
ejpam-5887	383	89	r⌊nk	r⌊nk	VERB
ejpam-5887	383	90	⌋+	⌋+	X
ejpam-5887	383	91	1	1	NUM
ejpam-5887	383	92	;	;	PUNCT
ejpam-5887	383	93	0	0	NUM
ejpam-5887	383	94	≤	≤	NUM
ejpam-5887	383	95	i	i	PRON
ejpam-5887	383	96	≤	≤	PROPN
ejpam-5887	383	97	r.	r.	PROPN
ejpam-5887	383	98	clearly	clearly	ADV
ejpam-5887	383	99	,	,	PUNCT
ejpam-5887	383	100	|ef	|ef	PROPN
ejpam-5887	383	101	(	(	PUNCT
ejpam-5887	383	102	i	i	NOUN
ejpam-5887	383	103	)	)	PUNCT
ejpam-5887	383	104	−	−	PROPN
ejpam-5887	383	105	ef	ef	PROPN
ejpam-5887	383	106	(	(	PUNCT
ejpam-5887	383	107	j)|	j)|	PROPN
ejpam-5887	383	108	≤	≤	NUM
ejpam-5887	383	109	1	1	NUM
ejpam-5887	383	110	and	and	CCONJ
ejpam-5887	383	111	|vf∗(i	|vf∗(i	NUM
ejpam-5887	383	112	)	)	PUNCT
ejpam-5887	383	113	−	−	NOUN
ejpam-5887	383	114	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	383	115	≤	≤	NOUN
ejpam-5887	383	116	1	1	NUM
ejpam-5887	383	117	for	for	ADP
ejpam-5887	383	118	i	i	PRON
ejpam-5887	383	119	,	,	PUNCT
ejpam-5887	383	120	j	j	PROPN
ejpam-5887	383	121	∈	∈	PROPN
ejpam-5887	383	122	{	{	PUNCT
ejpam-5887	383	123	0	0	NUM
ejpam-5887	383	124	,	,	PUNCT
ejpam-5887	383	125	1	1	NUM
ejpam-5887	383	126	,	,	PUNCT
ejpam-5887	383	127	...	...	PUNCT
ejpam-5887	383	128	,	,	PUNCT
ejpam-5887	383	129	k	k	PROPN
ejpam-5887	383	130	−	−	PROPN
ejpam-5887	383	131	1	1	NUM
ejpam-5887	383	132	}	}	PUNCT
ejpam-5887	383	133	.	.	PUNCT
ejpam-5887	384	1	hence	hence	ADV
ejpam-5887	384	2	,	,	PUNCT
ejpam-5887	384	3	p	p	X
ejpam-5887	384	4	(	(	PUNCT
ejpam-5887	384	5	n.kv	n.kv	PROPN
ejpam-5887	384	6	1,m	1,m	NOUN
ejpam-5887	384	7	)	)	PUNCT
ejpam-5887	384	8	is	be	AUX
ejpam-5887	384	9	an	an	DET
ejpam-5887	384	10	edge	edge	NOUN
ejpam-5887	384	11	k	k	NOUN
ejpam-5887	384	12	-	-	PUNCT
ejpam-5887	384	13	product	product	NOUN
ejpam-5887	384	14	cordial	cordial	ADJ
ejpam-5887	384	15	graph	graph	NOUN
ejpam-5887	384	16	if	if	SCONJ
ejpam-5887	384	17	n	n	PRON
ejpam-5887	384	18	≡	≡	PROPN
ejpam-5887	384	19	0	0	NUM
ejpam-5887	384	20	,	,	PUNCT
ejpam-5887	384	21	1	1	NUM
ejpam-5887	384	22	(	(	PUNCT
ejpam-5887	384	23	mod	mod	NOUN
ejpam-5887	384	24	k	k	PROPN
ejpam-5887	384	25	)	)	PUNCT
ejpam-5887	384	26	.	.	PUNCT
ejpam-5887	385	1	example	example	NOUN
ejpam-5887	386	1	3	3	NUM
ejpam-5887	386	2	.	.	PUNCT
ejpam-5887	386	3	an	an	DET
ejpam-5887	386	4	edge	edge	NOUN
ejpam-5887	386	5	4	4	NUM
ejpam-5887	386	6	-	-	PUNCT
ejpam-5887	386	7	product	product	NOUN
ejpam-5887	386	8	cordial	cordial	ADJ
ejpam-5887	386	9	labeling	labeling	NOUN
ejpam-5887	386	10	of	of	ADP
ejpam-5887	386	11	p	p	NOUN
ejpam-5887	386	12	(	(	PUNCT
ejpam-5887	386	13	4.kv	4.kv	NOUN
ejpam-5887	386	14	1,5	1,5	NUM
ejpam-5887	386	15	)	)	PUNCT
ejpam-5887	386	16	is	be	AUX
ejpam-5887	386	17	shown	show	VERB
ejpam-5887	386	18	in	in	ADP
ejpam-5887	386	19	figure	figure	NOUN
ejpam-5887	386	20	3	3	NUM
ejpam-5887	386	21	.	.	PUNCT
ejpam-5887	386	22	n.	n.	PROPN
ejpam-5887	386	23	m.	m.	PROPN
ejpam-5887	387	1	noureldeen	noureldeen	INTJ
ejpam-5887	387	2	et	et	PROPN
ejpam-5887	387	3	al	al	PROPN
ejpam-5887	387	4	.	.	PUNCT
ejpam-5887	387	5	/	/	SYM
ejpam-5887	387	6	eur	eur	PROPN
ejpam-5887	387	7	.	.	PUNCT
ejpam-5887	388	1	j.	j.	PROPN
ejpam-5887	388	2	pure	pure	PROPN
ejpam-5887	388	3	appl	appl	PROPN
ejpam-5887	388	4	.	.	PROPN
ejpam-5887	388	5	math	math	PROPN
ejpam-5887	388	6	,	,	PUNCT
ejpam-5887	388	7	18	18	NUM
ejpam-5887	388	8	(	(	PUNCT
ejpam-5887	388	9	2	2	NUM
ejpam-5887	388	10	)	)	PUNCT
ejpam-5887	388	11	(	(	PUNCT
ejpam-5887	388	12	2025	2025	NUM
ejpam-5887	388	13	)	)	PUNCT
ejpam-5887	388	14	,	,	PUNCT
ejpam-5887	388	15	5887	5887	NUM
ejpam-5887	388	16	12	12	NUM
ejpam-5887	388	17	of	of	ADP
ejpam-5887	388	18	21	21	NUM
ejpam-5887	388	19	0	0	NUM
ejpam-5887	388	20	0	0	NUM
ejpam-5887	388	21	0	0	NUM
ejpam-5887	388	22	0	0	NUM
ejpam-5887	388	23	0	0	NUM
ejpam-5887	388	24	0	0	NUM
ejpam-5887	388	25	0	0	NUM
ejpam-5887	388	26	0	0	NUM
ejpam-5887	388	27	0	0	NUM
ejpam-5887	388	28	0	0	NUM
ejpam-5887	388	29	0	0	NUM
ejpam-5887	388	30	1	1	NUM
ejpam-5887	388	31	1	1	NUM
ejpam-5887	388	32	1	1	NUM
ejpam-5887	388	33	1	1	NUM
ejpam-5887	388	34	1	1	NUM
ejpam-5887	388	35	1	1	NUM
ejpam-5887	388	36	1	1	NUM
ejpam-5887	388	37	1	1	NUM
ejpam-5887	388	38	1	1	NUM
ejpam-5887	388	39	1	1	NUM
ejpam-5887	388	40	1	1	NUM
ejpam-5887	388	41	12	12	NUM
ejpam-5887	388	42	2	2	NUM
ejpam-5887	388	43	2	2	NUM
ejpam-5887	388	44	2	2	NUM
ejpam-5887	388	45	2	2	NUM
ejpam-5887	388	46	2	2	NUM
ejpam-5887	388	47	2	2	NUM
ejpam-5887	388	48	2	2	NUM
ejpam-5887	388	49	2	2	NUM
ejpam-5887	388	50	2	2	NUM
ejpam-5887	388	51	2	2	NUM
ejpam-5887	388	52	23	23	NUM
ejpam-5887	388	53	3	3	NUM
ejpam-5887	388	54	3	3	NUM
ejpam-5887	388	55	3	3	NUM
ejpam-5887	388	56	3	3	NUM
ejpam-5887	388	57	3	3	NUM
ejpam-5887	388	58	3	3	NUM
ejpam-5887	388	59	3	3	NUM
ejpam-5887	388	60	3	3	NUM
ejpam-5887	388	61	3	3	NUM
ejpam-5887	388	62	3	3	NUM
ejpam-5887	388	63	3	3	NUM
ejpam-5887	388	64	figure	figure	NOUN
ejpam-5887	388	65	3	3	NUM
ejpam-5887	388	66	:	:	PUNCT
ejpam-5887	388	67	edge	edge	VERB
ejpam-5887	388	68	4	4	NUM
ejpam-5887	388	69	-	-	PUNCT
ejpam-5887	388	70	product	product	NOUN
ejpam-5887	388	71	cordial	cordial	ADJ
ejpam-5887	388	72	labeling	labeling	NOUN
ejpam-5887	388	73	of	of	ADP
ejpam-5887	388	74	p	p	NOUN
ejpam-5887	388	75	(	(	PUNCT
ejpam-5887	388	76	4.kv	4.kv	NOUN
ejpam-5887	388	77	1,5	1,5	NUM
ejpam-5887	388	78	)	)	PUNCT
ejpam-5887	388	79	theorem	theorem	VERB
ejpam-5887	388	80	16	16	NUM
ejpam-5887	388	81	.	.	PUNCT
ejpam-5887	389	1	the	the	DET
ejpam-5887	389	2	path	path	PROPN
ejpam-5887	389	3	union	union	PROPN
ejpam-5887	389	4	star	star	PROPN
ejpam-5887	389	5	graph	graph	NOUN
ejpam-5887	389	6	p	p	X
ejpam-5887	389	7	(	(	PUNCT
ejpam-5887	389	8	n.kv	n.kv	PROPN
ejpam-5887	389	9	1,m	1,m	NOUN
ejpam-5887	389	10	)	)	PUNCT
ejpam-5887	389	11	,	,	PUNCT
ejpam-5887	389	12	where	where	SCONJ
ejpam-5887	389	13	v	v	NOUN
ejpam-5887	389	14	is	be	AUX
ejpam-5887	389	15	a	a	DET
ejpam-5887	389	16	root	root	NOUN
ejpam-5887	389	17	vertex	vertex	NOUN
ejpam-5887	389	18	of	of	ADP
ejpam-5887	389	19	k1,m	k1,m	PROPN
ejpam-5887	389	20	admits	admit	VERB
ejpam-5887	389	21	an	an	DET
ejpam-5887	389	22	edge	edge	NOUN
ejpam-5887	389	23	k	k	NOUN
ejpam-5887	389	24	-	-	PUNCT
ejpam-5887	389	25	product	product	NOUN
ejpam-5887	389	26	cordial	cordial	ADJ
ejpam-5887	389	27	labeling	labeling	NOUN
ejpam-5887	389	28	if	if	SCONJ
ejpam-5887	389	29	n	n	PRON
ejpam-5887	389	30	≡	≡	PROPN
ejpam-5887	389	31	k−	k−	PROPN
ejpam-5887	389	32	1	1	NUM
ejpam-5887	389	33	(	(	PUNCT
ejpam-5887	389	34	mod	mod	NOUN
ejpam-5887	389	35	k	k	PROPN
ejpam-5887	389	36	)	)	PUNCT
ejpam-5887	389	37	and	and	CCONJ
ejpam-5887	389	38	m	m	PROPN
ejpam-5887	389	39	≡	≡	PROPN
ejpam-5887	389	40	0	0	NUM
ejpam-5887	389	41	,	,	PUNCT
ejpam-5887	389	42	k−	k−	NOUN
ejpam-5887	389	43	1	1	NUM
ejpam-5887	389	44	(	(	PUNCT
ejpam-5887	389	45	mod	mod	PROPN
ejpam-5887	389	46	k	k	PROPN
ejpam-5887	389	47	)	)	PUNCT
ejpam-5887	389	48	.	.	PUNCT
ejpam-5887	390	1	proof	proof	NOUN
ejpam-5887	390	2	.	.	PUNCT
ejpam-5887	391	1	let	let	VERB
ejpam-5887	391	2	the	the	DET
ejpam-5887	391	3	vertex	vertex	NOUN
ejpam-5887	391	4	and	and	CCONJ
ejpam-5887	391	5	edge	edge	NOUN
ejpam-5887	391	6	set	set	NOUN
ejpam-5887	391	7	of	of	ADP
ejpam-5887	391	8	p	p	X
ejpam-5887	391	9	(	(	PUNCT
ejpam-5887	391	10	n.kv	n.kv	PROPN
ejpam-5887	391	11	1,m	1,m	NOUN
ejpam-5887	391	12	)	)	PUNCT
ejpam-5887	391	13	be	be	AUX
ejpam-5887	391	14	v	v	PRON
ejpam-5887	391	15	(	(	PUNCT
ejpam-5887	391	16	p	p	X
ejpam-5887	391	17	(	(	PUNCT
ejpam-5887	391	18	n.kv	n.kv	PROPN
ejpam-5887	391	19	1,m	1,m	NOUN
ejpam-5887	391	20	)	)	PUNCT
ejpam-5887	391	21	)	)	PUNCT
ejpam-5887	392	1	=	=	PRON
ejpam-5887	392	2	{	{	PUNCT
ejpam-5887	392	3	vi	vi	PROPN
ejpam-5887	392	4	,	,	PUNCT
ejpam-5887	392	5	vji	vji	NOUN
ejpam-5887	392	6	:	:	PUNCT
ejpam-5887	392	7	1	1	NUM
ejpam-5887	392	8	≤	≤	NUM
ejpam-5887	392	9	i	i	PRON
ejpam-5887	392	10	≤	≤	PROPN
ejpam-5887	392	11	n	n	CCONJ
ejpam-5887	392	12	,	,	PUNCT
ejpam-5887	392	13	1	1	NUM
ejpam-5887	392	14	≤	≤	NUM
ejpam-5887	393	1	j	j	PROPN
ejpam-5887	393	2	≤	≤	PROPN
ejpam-5887	393	3	m	m	PROPN
ejpam-5887	393	4	}	}	PUNCT
ejpam-5887	393	5	and	and	CCONJ
ejpam-5887	393	6	e(p	e(p	PROPN
ejpam-5887	393	7	(	(	PUNCT
ejpam-5887	393	8	n.kv	n.kv	PROPN
ejpam-5887	393	9	1,m	1,m	NOUN
ejpam-5887	393	10	)	)	PUNCT
ejpam-5887	393	11	)	)	PUNCT
ejpam-5887	394	1	=	=	PRON
ejpam-5887	394	2	{	{	PUNCT
ejpam-5887	394	3	vivi+1	vivi+1	PROPN
ejpam-5887	394	4	,	,	PUNCT
ejpam-5887	394	5	viv	viv	PROPN
ejpam-5887	394	6	j	j	PROPN
ejpam-5887	395	1	i	i	PROPN
ejpam-5887	395	2	,	,	PUNCT
ejpam-5887	395	3	vnv	vnv	NOUN
ejpam-5887	395	4	j	j	PROPN
ejpam-5887	395	5	n	n	CCONJ
ejpam-5887	395	6	:	:	PUNCT
ejpam-5887	395	7	1	1	NUM
ejpam-5887	395	8	≤	≤	NUM
ejpam-5887	395	9	i	i	PRON
ejpam-5887	395	10	≤	≤	ADJ
ejpam-5887	395	11	n	n	CCONJ
ejpam-5887	395	12	−	−	PROPN
ejpam-5887	395	13	1	1	NUM
ejpam-5887	395	14	,	,	PUNCT
ejpam-5887	395	15	1	1	NUM
ejpam-5887	395	16	≤	≤	NUM
ejpam-5887	395	17	j	j	PROPN
ejpam-5887	395	18	≤	≤	PROPN
ejpam-5887	395	19	m	m	VERB
ejpam-5887	395	20	}	}	PUNCT
ejpam-5887	395	21	respectively	respectively	ADV
ejpam-5887	395	22	.	.	PUNCT
ejpam-5887	396	1	if	if	SCONJ
ejpam-5887	396	2	m	m	VERB
ejpam-5887	396	3	≡	≡	PROPN
ejpam-5887	396	4	0	0	PUNCT
ejpam-5887	397	1	(	(	PUNCT
ejpam-5887	397	2	mod	mod	PROPN
ejpam-5887	397	3	k	k	PROPN
ejpam-5887	397	4	)	)	PUNCT
ejpam-5887	397	5	,	,	PUNCT
ejpam-5887	397	6	then	then	ADV
ejpam-5887	397	7	by	by	ADP
ejpam-5887	397	8	theorems	theorem	NOUN
ejpam-5887	397	9	1	1	NUM
ejpam-5887	397	10	and	and	CCONJ
ejpam-5887	397	11	14	14	NUM
ejpam-5887	397	12	,	,	PUNCT
ejpam-5887	397	13	p	p	X
ejpam-5887	397	14	(	(	PUNCT
ejpam-5887	397	15	n.kv	n.kv	PROPN
ejpam-5887	397	16	1,m	1,m	NOUN
ejpam-5887	397	17	)	)	PUNCT
ejpam-5887	397	18	is	be	AUX
ejpam-5887	397	19	an	an	DET
ejpam-5887	397	20	edge	edge	NOUN
ejpam-5887	397	21	k	k	NOUN
ejpam-5887	397	22	-	-	PUNCT
ejpam-5887	397	23	product	product	NOUN
ejpam-5887	397	24	cordial	cordial	ADJ
ejpam-5887	397	25	graph	graph	NOUN
ejpam-5887	397	26	.	.	PUNCT
ejpam-5887	398	1	define	define	VERB
ejpam-5887	398	2	f	f	PROPN
ejpam-5887	398	3	:	:	PUNCT
ejpam-5887	398	4	e(p	e(p	PROPN
ejpam-5887	398	5	(	(	PUNCT
ejpam-5887	398	6	n.kv	n.kv	PROPN
ejpam-5887	398	7	1,m	1,m	NOUN
ejpam-5887	398	8	)	)	PUNCT
ejpam-5887	398	9	)	)	PUNCT
ejpam-5887	399	1	→	→	PUNCT
ejpam-5887	399	2	{	{	PUNCT
ejpam-5887	399	3	0	0	NUM
ejpam-5887	399	4	,	,	PUNCT
ejpam-5887	399	5	1	1	NUM
ejpam-5887	399	6	,	,	PUNCT
ejpam-5887	399	7	2	2	NUM
ejpam-5887	399	8	,	,	PUNCT
ejpam-5887	399	9	...	...	PUNCT
ejpam-5887	399	10	,	,	PUNCT
ejpam-5887	399	11	k−1	k−1	PROPN
ejpam-5887	399	12	}	}	PUNCT
ejpam-5887	399	13	for	for	ADP
ejpam-5887	399	14	n	n	DET
ejpam-5887	399	15	≡	≡	PROPN
ejpam-5887	399	16	k−1	k−1	PROPN
ejpam-5887	399	17	(	(	PUNCT
ejpam-5887	399	18	mod	mod	PROPN
ejpam-5887	399	19	k	k	PROPN
ejpam-5887	399	20	)	)	PUNCT
ejpam-5887	399	21	and	and	CCONJ
ejpam-5887	399	22	m	m	PROPN
ejpam-5887	399	23	≡	≡	PROPN
ejpam-5887	399	24	k−1	k−1	PROPN
ejpam-5887	399	25	(	(	PUNCT
ejpam-5887	399	26	mod	mod	PROPN
ejpam-5887	399	27	k	k	PROPN
ejpam-5887	399	28	)	)	PUNCT
ejpam-5887	399	29	as	as	SCONJ
ejpam-5887	399	30	follows	follow	VERB
ejpam-5887	399	31	:	:	PUNCT
ejpam-5887	399	32	f(viv	f(viv	PROPN
ejpam-5887	399	33	j	j	PROPN
ejpam-5887	399	34	i	i	PROPN
ejpam-5887	399	35	)	)	PUNCT
ejpam-5887	399	36	for	for	ADP
ejpam-5887	399	37	1	1	NUM
ejpam-5887	399	38	≤	≤	NUM
ejpam-5887	399	39	i	i	PRON
ejpam-5887	399	40	≤	≤	ADJ
ejpam-5887	400	1	n−	n−	NOUN
ejpam-5887	400	2	k	k	PROPN
ejpam-5887	401	1	+	+	CCONJ
ejpam-5887	401	2	1	1	NUM
ejpam-5887	401	3	,	,	PUNCT
ejpam-5887	401	4	1	1	NUM
ejpam-5887	401	5	≤	≤	NUM
ejpam-5887	401	6	j	j	PROPN
ejpam-5887	401	7	≤	≤	NUM
ejpam-5887	401	8	m	m	VERB
ejpam-5887	401	9	as	as	ADP
ejpam-5887	401	10	in	in	ADP
ejpam-5887	401	11	case	case	NOUN
ejpam-5887	401	12	(	(	PUNCT
ejpam-5887	401	13	i	i	NOUN
ejpam-5887	401	14	)	)	PUNCT
ejpam-5887	401	15	of	of	ADP
ejpam-5887	401	16	theorem	theorem	ADJ
ejpam-5887	401	17	15	15	NUM
ejpam-5887	401	18	,	,	PUNCT
ejpam-5887	401	19	f(vivi+1	f(vivi+1	NOUN
ejpam-5887	401	20	)	)	PUNCT
ejpam-5887	401	21	=	=	SYM
ejpam-5887	401	22	0	0	NUM
ejpam-5887	401	23	;	;	PUNCT
ejpam-5887	401	24	1	1	NUM
ejpam-5887	401	25	≤	≤	NUM
ejpam-5887	402	1	i	i	PRON
ejpam-5887	402	2	≤	≤	ADJ
ejpam-5887	402	3	n−	n−	PROPN
ejpam-5887	402	4	1	1	NUM
ejpam-5887	402	5	,	,	PUNCT
ejpam-5887	402	6	f(viv	f(viv	PROPN
ejpam-5887	402	7	j	j	PROPN
ejpam-5887	403	1	i	i	NOUN
ejpam-5887	403	2	)	)	PUNCT
ejpam-5887	404	1	=	=	PUNCT
ejpam-5887	404	2			NOUN
ejpam-5887	404	3	0	0	NUM
ejpam-5887	404	4	;	;	PUNCT
ejpam-5887	404	5	n−	n−	NOUN
ejpam-5887	404	6	k	k	NOUN
ejpam-5887	405	1	+	+	CCONJ
ejpam-5887	405	2	2	2	X
ejpam-5887	405	3	≤	≤	NUM
ejpam-5887	405	4	i	i	PRON
ejpam-5887	405	5	≤	≤	ADJ
ejpam-5887	405	6	n	n	CCONJ
ejpam-5887	405	7	,	,	PUNCT
ejpam-5887	405	8	1	1	NUM
ejpam-5887	405	9	≤	≤	NUM
ejpam-5887	405	10	j	j	PROPN
ejpam-5887	405	11	≤	≤	PROPN
ejpam-5887	405	12	⌊mk	⌊mk	PROPN
ejpam-5887	405	13	⌋	⌋	NOUN
ejpam-5887	405	14	1	1	NUM
ejpam-5887	405	15	;	;	PUNCT
ejpam-5887	405	16	n−	n−	NOUN
ejpam-5887	405	17	k	k	NOUN
ejpam-5887	406	1	+	+	CCONJ
ejpam-5887	406	2	2	2	X
ejpam-5887	406	3	≤	≤	NUM
ejpam-5887	406	4	i	i	PRON
ejpam-5887	406	5	≤	≤	ADJ
ejpam-5887	406	6	n	n	CCONJ
ejpam-5887	406	7	,	,	PUNCT
ejpam-5887	406	8	⌊mk	⌊mk	NOUN
ejpam-5887	406	9	⌋+	⌋+	PUNCT
ejpam-5887	406	10	1	1	NUM
ejpam-5887	406	11	≤	≤	NUM
ejpam-5887	406	12	j	j	PROPN
ejpam-5887	406	13	≤	≤	PROPN
ejpam-5887	406	14	2⌊mk	2⌊mk	NUM
ejpam-5887	406	15	⌋	⌋	NOUN
ejpam-5887	406	16	2	2	NUM
ejpam-5887	406	17	;	;	PUNCT
ejpam-5887	406	18	n−	n−	NOUN
ejpam-5887	406	19	k	k	NOUN
ejpam-5887	406	20	+	+	CCONJ
ejpam-5887	406	21	2	2	X
ejpam-5887	406	22	≤	≤	NUM
ejpam-5887	406	23	i	i	PRON
ejpam-5887	406	24	≤	≤	ADJ
ejpam-5887	406	25	n	n	CCONJ
ejpam-5887	406	26	,	,	PUNCT
ejpam-5887	406	27	2⌊mk	2⌊mk	NUM
ejpam-5887	406	28	⌋+	⌋+	NUM
ejpam-5887	406	29	1	1	NUM
ejpam-5887	406	30	≤	≤	NUM
ejpam-5887	406	31	j	j	PROPN
ejpam-5887	406	32	≤	≤	PROPN
ejpam-5887	406	33	3⌊mk	3⌊mk	NUM
ejpam-5887	406	34	⌋	⌋	NOUN
ejpam-5887	406	35	:	:	PUNCT
ejpam-5887	406	36	:	:	PUNCT
ejpam-5887	407	1	k	k	X
ejpam-5887	407	2	−	−	PROPN
ejpam-5887	407	3	1	1	NUM
ejpam-5887	407	4	;	;	PUNCT
ejpam-5887	407	5	n−	n−	NOUN
ejpam-5887	407	6	k	k	NOUN
ejpam-5887	407	7	+	+	CCONJ
ejpam-5887	407	8	2	2	X
ejpam-5887	407	9	≤	≤	NUM
ejpam-5887	407	10	i	i	PRON
ejpam-5887	407	11	≤	≤	ADJ
ejpam-5887	407	12	n	n	CCONJ
ejpam-5887	407	13	,	,	PUNCT
ejpam-5887	407	14	(	(	PUNCT
ejpam-5887	407	15	k	k	PROPN
ejpam-5887	407	16	−	−	PROPN
ejpam-5887	407	17	1)⌊mk	1)⌊mk	NUM
ejpam-5887	407	18	⌋+	⌋+	NUM
ejpam-5887	407	19	1	1	NUM
ejpam-5887	407	20	≤	≤	NUM
ejpam-5887	407	21	j	j	PROPN
ejpam-5887	407	22	≤	≤	PROPN
ejpam-5887	407	23	k⌊mk	k⌊mk	NOUN
ejpam-5887	407	24	⌋	⌋	NOUN
ejpam-5887	407	25	;	;	PUNCT
ejpam-5887	407	26	i	i	PRON
ejpam-5887	407	27	=	=	VERB
ejpam-5887	407	28	n−	n−	NOUN
ejpam-5887	407	29	k	k	NOUN
ejpam-5887	407	30	+	+	CCONJ
ejpam-5887	407	31	2	2	NUM
ejpam-5887	407	32	,	,	PUNCT
ejpam-5887	407	33	m−	m−	PROPN
ejpam-5887	407	34	k	k	PROPN
ejpam-5887	407	35	+	+	CCONJ
ejpam-5887	407	36	2	2	NUM
ejpam-5887	407	37	≤	≤	NUM
ejpam-5887	407	38	j	j	PROPN
ejpam-5887	407	39	≤	≤	NOUN
ejpam-5887	407	40	m	m	VERB
ejpam-5887	407	41	;	;	PUNCT
ejpam-5887	407	42	i	i	PRON
ejpam-5887	407	43	=	=	VERB
ejpam-5887	407	44	n−	n−	NOUN
ejpam-5887	407	45	k	k	NOUN
ejpam-5887	407	46	+	+	CCONJ
ejpam-5887	407	47	3	3	NUM
ejpam-5887	407	48	,	,	PUNCT
ejpam-5887	407	49	m−	m−	PROPN
ejpam-5887	407	50	k	k	PROPN
ejpam-5887	407	51	+	+	CCONJ
ejpam-5887	407	52	2	2	NUM
ejpam-5887	407	53	≤	≤	NUM
ejpam-5887	407	54	j	j	PROPN
ejpam-5887	407	55	≤	≤	NOUN
ejpam-5887	407	56	m	m	VERB
ejpam-5887	407	57	;	;	PUNCT
ejpam-5887	407	58	i	i	PRON
ejpam-5887	407	59	=	=	SYM
ejpam-5887	407	60	n	n	PROPN
ejpam-5887	407	61	,	,	PUNCT
ejpam-5887	407	62	m−	m−	PROPN
ejpam-5887	407	63	k	k	PROPN
ejpam-5887	407	64	+	+	CCONJ
ejpam-5887	407	65	2	2	NUM
ejpam-5887	407	66	≤	≤	NUM
ejpam-5887	407	67	j	j	PROPN
ejpam-5887	407	68	≤	≤	PROPN
ejpam-5887	407	69	m.	m.	NOUN
ejpam-5887	407	70	from	from	ADP
ejpam-5887	407	71	this	this	DET
ejpam-5887	407	72	labeling	labeling	NOUN
ejpam-5887	407	73	we	we	PRON
ejpam-5887	407	74	have	have	VERB
ejpam-5887	407	75	,	,	PUNCT
ejpam-5887	407	76	ef	ef	PROPN
ejpam-5887	407	77	(	(	PUNCT
ejpam-5887	407	78	i	i	NOUN
ejpam-5887	407	79	)	)	PUNCT
ejpam-5887	407	80	=	=	PRON
ejpam-5887	407	81	{	{	PUNCT
ejpam-5887	407	82	k⌊nk	k⌊nk	NOUN
ejpam-5887	407	83	⌋⌊	⌋⌊	SYM
ejpam-5887	407	84	m	m	VERB
ejpam-5887	407	85	k	k	ADJ
ejpam-5887	407	86	⌋+	⌋+	X
ejpam-5887	407	87	k⌊nk	k⌊nk	NOUN
ejpam-5887	407	88	⌋+	⌋+	PUNCT
ejpam-5887	407	89	(	(	PUNCT
ejpam-5887	407	90	k	k	NOUN
ejpam-5887	407	91	−	−	PROPN
ejpam-5887	407	92	1)(1	1)(1	NUM
ejpam-5887	407	93	+	+	CCONJ
ejpam-5887	407	94	⌊mk	⌊mk	X
ejpam-5887	407	95	⌋)−	⌋)−	SYM
ejpam-5887	407	96	1	1	NUM
ejpam-5887	407	97	;	;	PUNCT
ejpam-5887	407	98	i	i	PRON
ejpam-5887	407	99	=	=	SYM
ejpam-5887	407	100	0	0	NUM
ejpam-5887	407	101	k⌊nk	k⌊nk	NOUN
ejpam-5887	407	102	⌋⌊	⌋⌊	SYM
ejpam-5887	407	103	m	m	VERB
ejpam-5887	407	104	k	k	PRON
ejpam-5887	407	105	⌋+	⌋+	X
ejpam-5887	407	106	k⌊nk	k⌊nk	NOUN
ejpam-5887	407	107	⌋+	⌋+	PUNCT
ejpam-5887	407	108	(	(	PUNCT
ejpam-5887	407	109	k	k	NOUN
ejpam-5887	407	110	−	−	PROPN
ejpam-5887	407	111	1)(1	1)(1	NUM
ejpam-5887	407	112	+	+	CCONJ
ejpam-5887	407	113	⌊mk	⌊mk	NOUN
ejpam-5887	407	114	⌋	⌋	NOUN
ejpam-5887	407	115	)	)	PUNCT
ejpam-5887	407	116	;	;	PUNCT
ejpam-5887	407	117	1	1	NUM
ejpam-5887	407	118	≤	≤	NUM
ejpam-5887	407	119	i	i	X
ejpam-5887	407	120	≤	≤	NOUN
ejpam-5887	408	1	k	k	PRON
ejpam-5887	408	2	−	−	PROPN
ejpam-5887	408	3	1	1	NUM
ejpam-5887	408	4	,	,	PUNCT
ejpam-5887	408	5	vf∗(i	vf∗(i	ADJ
ejpam-5887	408	6	)	)	PUNCT
ejpam-5887	408	7	=	=	SYM
ejpam-5887	408	8	k⌊nk	k⌊nk	NOUN
ejpam-5887	408	9	⌋⌊	⌋⌊	SYM
ejpam-5887	408	10	m	m	VERB
ejpam-5887	408	11	k	k	ADJ
ejpam-5887	408	12	⌋+	⌋+	X
ejpam-5887	408	13	k⌊nk	k⌊nk	NOUN
ejpam-5887	408	14	⌋+	⌋+	PUNCT
ejpam-5887	408	15	(	(	PUNCT
ejpam-5887	408	16	k	k	NOUN
ejpam-5887	408	17	−	−	PROPN
ejpam-5887	408	18	1)(1	1)(1	NUM
ejpam-5887	408	19	+	+	CCONJ
ejpam-5887	408	20	⌊mk	⌊mk	NOUN
ejpam-5887	408	21	⌋	⌋	NOUN
ejpam-5887	408	22	)	)	PUNCT
ejpam-5887	408	23	;	;	PUNCT
ejpam-5887	408	24	0	0	NUM
ejpam-5887	408	25	≤	≤	NUM
ejpam-5887	408	26	i	i	PRON
ejpam-5887	408	27	≤	≤	NOUN
ejpam-5887	409	1	k	k	PRON
ejpam-5887	410	1	−	−	NOUN
ejpam-5887	410	2	1	1	X
ejpam-5887	410	3	.	.	PUNCT
ejpam-5887	411	1	clearly	clearly	ADV
ejpam-5887	411	2	,	,	PUNCT
ejpam-5887	411	3	|ef	|ef	PROPN
ejpam-5887	411	4	(	(	PUNCT
ejpam-5887	411	5	i	i	NOUN
ejpam-5887	411	6	)	)	PUNCT
ejpam-5887	411	7	−	−	PROPN
ejpam-5887	411	8	ef	ef	PROPN
ejpam-5887	411	9	(	(	PUNCT
ejpam-5887	411	10	j)|	j)|	PROPN
ejpam-5887	411	11	≤	≤	NUM
ejpam-5887	411	12	1	1	NUM
ejpam-5887	411	13	and	and	CCONJ
ejpam-5887	411	14	|vf∗(i	|vf∗(i	NUM
ejpam-5887	411	15	)	)	PUNCT
ejpam-5887	411	16	−	−	NOUN
ejpam-5887	411	17	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	411	18	≤	≤	NOUN
ejpam-5887	411	19	1	1	NUM
ejpam-5887	411	20	for	for	ADP
ejpam-5887	411	21	i	i	PRON
ejpam-5887	411	22	,	,	PUNCT
ejpam-5887	411	23	j	j	PROPN
ejpam-5887	411	24	∈	∈	PROPN
ejpam-5887	411	25	{	{	PUNCT
ejpam-5887	411	26	0	0	NUM
ejpam-5887	411	27	,	,	PUNCT
ejpam-5887	411	28	1	1	NUM
ejpam-5887	411	29	,	,	PUNCT
ejpam-5887	411	30	...	...	PUNCT
ejpam-5887	411	31	,	,	PUNCT
ejpam-5887	411	32	k	k	PROPN
ejpam-5887	411	33	−	−	PROPN
ejpam-5887	411	34	1	1	NUM
ejpam-5887	411	35	}	}	PUNCT
ejpam-5887	411	36	.	.	PUNCT
ejpam-5887	412	1	hence	hence	ADV
ejpam-5887	412	2	,	,	PUNCT
ejpam-5887	412	3	p	p	X
ejpam-5887	412	4	(	(	PUNCT
ejpam-5887	412	5	n.kv	n.kv	PROPN
ejpam-5887	412	6	1,m	1,m	NOUN
ejpam-5887	412	7	)	)	PUNCT
ejpam-5887	412	8	is	be	AUX
ejpam-5887	412	9	an	an	DET
ejpam-5887	412	10	edge	edge	NOUN
ejpam-5887	412	11	k	k	NOUN
ejpam-5887	412	12	-	-	PUNCT
ejpam-5887	412	13	product	product	NOUN
ejpam-5887	412	14	cordial	cordial	ADJ
ejpam-5887	412	15	graph	graph	NOUN
ejpam-5887	412	16	if	if	SCONJ
ejpam-5887	412	17	n	n	PRON
ejpam-5887	412	18	≡	≡	PROPN
ejpam-5887	412	19	k−1	k−1	PROPN
ejpam-5887	412	20	(	(	PUNCT
ejpam-5887	412	21	mod	mod	PROPN
ejpam-5887	412	22	k	k	PROPN
ejpam-5887	412	23	)	)	PUNCT
ejpam-5887	412	24	andm	andm	PROPN
ejpam-5887	412	25	≡	≡	PROPN
ejpam-5887	412	26	0	0	PROPN
ejpam-5887	412	27	,	,	PUNCT
ejpam-5887	412	28	k−1	k−1	PROPN
ejpam-5887	412	29	(	(	PUNCT
ejpam-5887	412	30	mod	mod	PROPN
ejpam-5887	412	31	k	k	PROPN
ejpam-5887	412	32	)	)	PUNCT
ejpam-5887	412	33	.	.	PUNCT
ejpam-5887	413	1	example	example	NOUN
ejpam-5887	414	1	4	4	NUM
ejpam-5887	414	2	.	.	PUNCT
ejpam-5887	414	3	an	an	DET
ejpam-5887	414	4	edge	edge	NOUN
ejpam-5887	414	5	5	5	NUM
ejpam-5887	414	6	-	-	PUNCT
ejpam-5887	414	7	product	product	NOUN
ejpam-5887	414	8	cordial	cordial	ADJ
ejpam-5887	414	9	labeling	labeling	NOUN
ejpam-5887	414	10	of	of	ADP
ejpam-5887	414	11	p	p	NOUN
ejpam-5887	414	12	(	(	PUNCT
ejpam-5887	414	13	4.kv	4.kv	NOUN
ejpam-5887	414	14	1,5	1,5	NUM
ejpam-5887	414	15	)	)	PUNCT
ejpam-5887	414	16	is	be	AUX
ejpam-5887	414	17	shown	show	VERB
ejpam-5887	414	18	in	in	ADP
ejpam-5887	414	19	figure	figure	NOUN
ejpam-5887	414	20	4	4	NUM
ejpam-5887	414	21	.	.	PUNCT
ejpam-5887	415	1	n.	n.	PROPN
ejpam-5887	415	2	m.	m.	PROPN
ejpam-5887	415	3	noureldeen	noureldeen	INTJ
ejpam-5887	415	4	et	et	PROPN
ejpam-5887	415	5	al	al	PROPN
ejpam-5887	415	6	.	.	PUNCT
ejpam-5887	415	7	/	/	SYM
ejpam-5887	415	8	eur	eur	PROPN
ejpam-5887	415	9	.	.	PUNCT
ejpam-5887	416	1	j.	j.	PROPN
ejpam-5887	416	2	pure	pure	PROPN
ejpam-5887	416	3	appl	appl	PROPN
ejpam-5887	416	4	.	.	PROPN
ejpam-5887	416	5	math	math	PROPN
ejpam-5887	416	6	,	,	PUNCT
ejpam-5887	416	7	18	18	NUM
ejpam-5887	416	8	(	(	PUNCT
ejpam-5887	416	9	2	2	NUM
ejpam-5887	416	10	)	)	PUNCT
ejpam-5887	416	11	(	(	PUNCT
ejpam-5887	416	12	2025	2025	NUM
ejpam-5887	416	13	)	)	PUNCT
ejpam-5887	416	14	,	,	PUNCT
ejpam-5887	416	15	5887	5887	NUM
ejpam-5887	416	16	13	13	NUM
ejpam-5887	416	17	of	of	ADP
ejpam-5887	416	18	21	21	NUM
ejpam-5887	416	19	0	0	NUM
ejpam-5887	416	20	0	0	NUM
ejpam-5887	416	21	0	0	NUM
ejpam-5887	416	22	0	0	NUM
ejpam-5887	416	23	0	0	NUM
ejpam-5887	416	24	1	1	NUM
ejpam-5887	416	25	1	1	NUM
ejpam-5887	416	26	1	1	NUM
ejpam-5887	416	27	1	1	NUM
ejpam-5887	416	28	1	1	NUM
ejpam-5887	416	29	1	1	NUM
ejpam-5887	416	30	1	1	NUM
ejpam-5887	416	31	1	1	NUM
ejpam-5887	416	32	2	2	NUM
ejpam-5887	416	33	2	2	NUM
ejpam-5887	416	34	2	2	NUM
ejpam-5887	416	35	2	2	NUM
ejpam-5887	416	36	2	2	NUM
ejpam-5887	416	37	2	2	NUM
ejpam-5887	416	38	2	2	NUM
ejpam-5887	416	39	2	2	NUM
ejpam-5887	416	40	2	2	NUM
ejpam-5887	416	41	2	2	NUM
ejpam-5887	416	42	3	3	NUM
ejpam-5887	416	43	3	3	NUM
ejpam-5887	416	44	3	3	NUM
ejpam-5887	416	45	3	3	NUM
ejpam-5887	416	46	3	3	NUM
ejpam-5887	416	47	3	3	NUM
ejpam-5887	416	48	3	3	NUM
ejpam-5887	416	49	3	3	NUM
ejpam-5887	416	50	3	3	NUM
ejpam-5887	416	51	3	3	NUM
ejpam-5887	416	52	4	4	NUM
ejpam-5887	416	53	4	4	NUM
ejpam-5887	416	54	4	4	NUM
ejpam-5887	416	55	4	4	NUM
ejpam-5887	416	56	4	4	NUM
ejpam-5887	416	57	4	4	NUM
ejpam-5887	416	58	4	4	NUM
ejpam-5887	416	59	4	4	NUM
ejpam-5887	416	60	4	4	NUM
ejpam-5887	416	61	4	4	NUM
ejpam-5887	416	62	figure	figure	NOUN
ejpam-5887	416	63	4	4	NUM
ejpam-5887	416	64	:	:	PUNCT
ejpam-5887	416	65	edge	edge	VERB
ejpam-5887	416	66	5	5	NUM
ejpam-5887	416	67	-	-	PUNCT
ejpam-5887	416	68	product	product	NOUN
ejpam-5887	416	69	cordial	cordial	ADJ
ejpam-5887	416	70	labeling	labeling	NOUN
ejpam-5887	416	71	of	of	ADP
ejpam-5887	416	72	p	p	NOUN
ejpam-5887	416	73	(	(	PUNCT
ejpam-5887	416	74	4.kv	4.kv	NOUN
ejpam-5887	416	75	1,5	1,5	NUM
ejpam-5887	416	76	)	)	PUNCT
ejpam-5887	416	77	4.2	4.2	NUM
ejpam-5887	416	78	.	.	PUNCT
ejpam-5887	417	1	path	path	PROPN
ejpam-5887	417	2	union	union	PROPN
ejpam-5887	417	3	of	of	ADP
ejpam-5887	417	4	bistar	bistar	PROPN
ejpam-5887	417	5	in	in	ADP
ejpam-5887	417	6	this	this	DET
ejpam-5887	417	7	subsection	subsection	NOUN
ejpam-5887	417	8	,	,	PUNCT
ejpam-5887	417	9	we	we	PRON
ejpam-5887	417	10	prove	prove	VERB
ejpam-5887	417	11	that	that	SCONJ
ejpam-5887	417	12	the	the	DET
ejpam-5887	417	13	path	path	NOUN
ejpam-5887	417	14	union	union	NOUN
ejpam-5887	417	15	of	of	ADP
ejpam-5887	417	16	a	a	DET
ejpam-5887	417	17	bistar	bistar	NOUN
ejpam-5887	417	18	graph	graph	NOUN
ejpam-5887	417	19	p	p	PROPN
ejpam-5887	417	20	(	(	PUNCT
ejpam-5887	417	21	n.bv	n.bv	NOUN
ejpam-5887	417	22	m	m	PROPN
ejpam-5887	417	23	,	,	PUNCT
ejpam-5887	417	24	m	m	PROPN
ejpam-5887	417	25	)	)	PUNCT
ejpam-5887	417	26	,	,	PUNCT
ejpam-5887	417	27	where	where	SCONJ
ejpam-5887	417	28	v	v	NOUN
ejpam-5887	417	29	is	be	AUX
ejpam-5887	417	30	a	a	DET
ejpam-5887	417	31	root	root	NOUN
ejpam-5887	417	32	vertex	vertex	NOUN
ejpam-5887	417	33	of	of	ADP
ejpam-5887	417	34	bm	bm	PROPN
ejpam-5887	417	35	,	,	PUNCT
ejpam-5887	417	36	m	m	PROPN
ejpam-5887	417	37	admits	admit	VERB
ejpam-5887	417	38	an	an	DET
ejpam-5887	417	39	edge	edge	NOUN
ejpam-5887	417	40	k	k	NOUN
ejpam-5887	417	41	-	-	PUNCT
ejpam-5887	417	42	product	product	NOUN
ejpam-5887	417	43	cordial	cordial	ADJ
ejpam-5887	417	44	labeling	labeling	NOUN
ejpam-5887	417	45	for	for	ADP
ejpam-5887	417	46	n	n	X
ejpam-5887	417	47	≡	≡	PROPN
ejpam-5887	417	48	0	0	NUM
ejpam-5887	417	49	,	,	PUNCT
ejpam-5887	417	50	1	1	NUM
ejpam-5887	417	51	(	(	PUNCT
ejpam-5887	417	52	mod	mod	NOUN
ejpam-5887	417	53	k	k	PROPN
ejpam-5887	417	54	)	)	PUNCT
ejpam-5887	417	55	.	.	PUNCT
ejpam-5887	418	1	also	also	ADV
ejpam-5887	418	2	,	,	PUNCT
ejpam-5887	418	3	we	we	PRON
ejpam-5887	418	4	show	show	VERB
ejpam-5887	418	5	that	that	SCONJ
ejpam-5887	418	6	the	the	DET
ejpam-5887	418	7	graph	graph	NOUN
ejpam-5887	418	8	p	p	X
ejpam-5887	418	9	(	(	PUNCT
ejpam-5887	418	10	n.bv	n.bv	NOUN
ejpam-5887	418	11	m	m	PROPN
ejpam-5887	418	12	,	,	PUNCT
ejpam-5887	418	13	m	m	PROPN
ejpam-5887	418	14	)	)	PUNCT
ejpam-5887	418	15	admits	admit	VERB
ejpam-5887	418	16	an	an	DET
ejpam-5887	418	17	edge	edge	NOUN
ejpam-5887	418	18	k	k	NOUN
ejpam-5887	418	19	-	-	PUNCT
ejpam-5887	418	20	product	product	NOUN
ejpam-5887	418	21	cordial	cordial	ADJ
ejpam-5887	418	22	labeling	labeling	NOUN
ejpam-5887	418	23	for	for	ADP
ejpam-5887	418	24	n	n	PRON
ejpam-5887	418	25	≡	≡	PROPN
ejpam-5887	418	26	k	k	PROPN
ejpam-5887	419	1	−	−	PROPN
ejpam-5887	419	2	1	1	NUM
ejpam-5887	419	3	(	(	PUNCT
ejpam-5887	419	4	mod	mod	PROPN
ejpam-5887	419	5	k	k	PROPN
ejpam-5887	419	6	)	)	PUNCT
ejpam-5887	419	7	if	if	SCONJ
ejpam-5887	419	8	m	m	VERB
ejpam-5887	419	9	≡	≡	PROPN
ejpam-5887	419	10	0	0	NUM
ejpam-5887	419	11	,	,	PUNCT
ejpam-5887	419	12	k	k	PROPN
ejpam-5887	420	1	−	−	PROPN
ejpam-5887	420	2	1	1	NUM
ejpam-5887	420	3	(	(	PUNCT
ejpam-5887	420	4	mod	mod	PROPN
ejpam-5887	420	5	k	k	PROPN
ejpam-5887	420	6	)	)	PUNCT
ejpam-5887	420	7	.	.	PUNCT
ejpam-5887	421	1	theorem	theorem	VERB
ejpam-5887	421	2	17	17	NUM
ejpam-5887	421	3	.	.	PUNCT
ejpam-5887	422	1	the	the	DET
ejpam-5887	422	2	path	path	PROPN
ejpam-5887	422	3	union	union	PROPN
ejpam-5887	422	4	of	of	ADP
ejpam-5887	422	5	bistar	bistar	PROPN
ejpam-5887	422	6	graph	graph	NOUN
ejpam-5887	422	7	p	p	PROPN
ejpam-5887	422	8	(	(	PUNCT
ejpam-5887	422	9	n.bv	n.bv	NOUN
ejpam-5887	422	10	m	m	PROPN
ejpam-5887	422	11	,	,	PUNCT
ejpam-5887	422	12	m	m	PROPN
ejpam-5887	422	13	)	)	PUNCT
ejpam-5887	422	14	,	,	PUNCT
ejpam-5887	422	15	where	where	SCONJ
ejpam-5887	422	16	v	v	NOUN
ejpam-5887	422	17	is	be	AUX
ejpam-5887	422	18	a	a	DET
ejpam-5887	422	19	root	root	NOUN
ejpam-5887	422	20	vertex	vertex	NOUN
ejpam-5887	422	21	of	of	ADP
ejpam-5887	422	22	bm	bm	PROPN
ejpam-5887	422	23	,	,	PUNCT
ejpam-5887	422	24	m	m	PROPN
ejpam-5887	422	25	admits	admit	VERB
ejpam-5887	422	26	an	an	DET
ejpam-5887	422	27	edge	edge	NOUN
ejpam-5887	422	28	k	k	NOUN
ejpam-5887	422	29	-	-	PUNCT
ejpam-5887	422	30	product	product	NOUN
ejpam-5887	422	31	cordial	cordial	ADJ
ejpam-5887	422	32	labeling	labeling	NOUN
ejpam-5887	422	33	if	if	SCONJ
ejpam-5887	422	34	n	n	PRON
ejpam-5887	422	35	≡	≡	PROPN
ejpam-5887	422	36	0	0	NUM
ejpam-5887	422	37	,	,	PUNCT
ejpam-5887	422	38	1	1	NUM
ejpam-5887	422	39	(	(	PUNCT
ejpam-5887	422	40	mod	mod	NOUN
ejpam-5887	422	41	k	k	PROPN
ejpam-5887	422	42	)	)	PUNCT
ejpam-5887	422	43	.	.	PUNCT
ejpam-5887	423	1	proof	proof	NOUN
ejpam-5887	423	2	.	.	PUNCT
ejpam-5887	424	1	let	let	VERB
ejpam-5887	424	2	the	the	DET
ejpam-5887	424	3	vertex	vertex	NOUN
ejpam-5887	424	4	and	and	CCONJ
ejpam-5887	424	5	edge	edge	NOUN
ejpam-5887	424	6	set	set	NOUN
ejpam-5887	424	7	of	of	ADP
ejpam-5887	424	8	p	p	PROPN
ejpam-5887	424	9	(	(	PUNCT
ejpam-5887	424	10	n.bv	n.bv	NOUN
ejpam-5887	424	11	m	m	PROPN
ejpam-5887	424	12	,	,	PUNCT
ejpam-5887	424	13	m	m	VERB
ejpam-5887	424	14	)	)	PUNCT
ejpam-5887	424	15	be	be	VERB
ejpam-5887	424	16	v	v	PRON
ejpam-5887	424	17	(	(	PUNCT
ejpam-5887	424	18	p	p	X
ejpam-5887	424	19	(	(	PUNCT
ejpam-5887	424	20	n.bv	n.bv	NOUN
ejpam-5887	424	21	m	m	PROPN
ejpam-5887	424	22	,	,	PUNCT
ejpam-5887	424	23	m	m	NOUN
ejpam-5887	424	24	)	)	PUNCT
ejpam-5887	424	25	)	)	PUNCT
ejpam-5887	425	1	=	=	PRON
ejpam-5887	425	2	{	{	PUNCT
ejpam-5887	425	3	vi	vi	PROPN
ejpam-5887	425	4	,	,	PUNCT
ejpam-5887	425	5	ui	ui	NOUN
ejpam-5887	425	6	,	,	PUNCT
ejpam-5887	425	7	vji	vji	VERB
ejpam-5887	425	8	,	,	PUNCT
ejpam-5887	425	9	u	u	NOUN
ejpam-5887	425	10	j	j	PROPN
ejpam-5887	426	1	i	i	PRON
ejpam-5887	426	2	:	:	PUNCT
ejpam-5887	426	3	1	1	NUM
ejpam-5887	426	4	≤	≤	NUM
ejpam-5887	426	5	i	i	PRON
ejpam-5887	426	6	≤	≤	PROPN
ejpam-5887	426	7	n	n	CCONJ
ejpam-5887	426	8	,	,	PUNCT
ejpam-5887	426	9	1	1	NUM
ejpam-5887	426	10	≤	≤	NUM
ejpam-5887	426	11	j	j	PROPN
ejpam-5887	426	12	≤	≤	PROPN
ejpam-5887	426	13	m	m	PROPN
ejpam-5887	426	14	}	}	PUNCT
ejpam-5887	426	15	and	and	CCONJ
ejpam-5887	426	16	e(p	e(p	PROPN
ejpam-5887	426	17	(	(	PUNCT
ejpam-5887	426	18	n.bv	n.bv	NOUN
ejpam-5887	426	19	m	m	PROPN
ejpam-5887	426	20	,	,	PUNCT
ejpam-5887	426	21	m	m	NOUN
ejpam-5887	426	22	)	)	PUNCT
ejpam-5887	426	23	)	)	PUNCT
ejpam-5887	427	1	=	=	PRON
ejpam-5887	427	2	{	{	PUNCT
ejpam-5887	427	3	vivi+1	vivi+1	PROPN
ejpam-5887	427	4	,	,	PUNCT
ejpam-5887	427	5	viui	viui	PROPN
ejpam-5887	427	6	,	,	PUNCT
ejpam-5887	427	7	viv	viv	PROPN
ejpam-5887	427	8	j	j	PROPN
ejpam-5887	427	9	i	i	PROPN
ejpam-5887	427	10	,	,	PUNCT
ejpam-5887	427	11	uiu	uiu	PROPN
ejpam-5887	427	12	j	j	PROPN
ejpam-5887	428	1	i	i	PROPN
ejpam-5887	428	2	,	,	PUNCT
ejpam-5887	428	3	vnun	vnun	PROPN
ejpam-5887	428	4	,	,	PUNCT
ejpam-5887	428	5	vnv	vnv	NOUN
ejpam-5887	428	6	j	j	PROPN
ejpam-5887	428	7	n	n	CCONJ
ejpam-5887	428	8	,	,	PUNCT
ejpam-5887	428	9	unu	unu	PROPN
ejpam-5887	428	10	j	j	PROPN
ejpam-5887	428	11	n	n	CCONJ
ejpam-5887	428	12	:	:	PUNCT
ejpam-5887	428	13	1	1	NUM
ejpam-5887	428	14	≤	≤	NUM
ejpam-5887	428	15	i	i	PRON
ejpam-5887	428	16	≤	≤	ADJ
ejpam-5887	428	17	n−	n−	PROPN
ejpam-5887	428	18	1	1	NUM
ejpam-5887	428	19	,	,	PUNCT
ejpam-5887	428	20	1	1	NUM
ejpam-5887	428	21	≤	≤	NUM
ejpam-5887	428	22	j	j	PROPN
ejpam-5887	428	23	≤	≤	PROPN
ejpam-5887	428	24	m	m	VERB
ejpam-5887	428	25	}	}	PUNCT
ejpam-5887	428	26	respectively	respectively	ADV
ejpam-5887	428	27	.	.	PUNCT
ejpam-5887	429	1	define	define	VERB
ejpam-5887	429	2	f	f	PROPN
ejpam-5887	429	3	:	:	PUNCT
ejpam-5887	429	4	e(p	e(p	PROPN
ejpam-5887	429	5	(	(	PUNCT
ejpam-5887	429	6	n.bv	n.bv	NOUN
ejpam-5887	429	7	m	m	PROPN
ejpam-5887	429	8	,	,	PUNCT
ejpam-5887	429	9	m	m	NOUN
ejpam-5887	429	10	)	)	PUNCT
ejpam-5887	429	11	)	)	PUNCT
ejpam-5887	430	1	→	→	PUNCT
ejpam-5887	430	2	{	{	PUNCT
ejpam-5887	430	3	0	0	NUM
ejpam-5887	430	4	,	,	PUNCT
ejpam-5887	430	5	1	1	NUM
ejpam-5887	430	6	,	,	PUNCT
ejpam-5887	430	7	2	2	NUM
ejpam-5887	430	8	,	,	PUNCT
ejpam-5887	430	9	...	...	PUNCT
ejpam-5887	430	10	,	,	PUNCT
ejpam-5887	430	11	k−1	k−1	PROPN
ejpam-5887	430	12	}	}	PUNCT
ejpam-5887	430	13	for	for	ADP
ejpam-5887	430	14	n	n	X
ejpam-5887	430	15	≡	≡	PROPN
ejpam-5887	430	16	0	0	NUM
ejpam-5887	430	17	,	,	PUNCT
ejpam-5887	430	18	1	1	NUM
ejpam-5887	430	19	(	(	PUNCT
ejpam-5887	430	20	mod	mod	PROPN
ejpam-5887	430	21	k	k	PROPN
ejpam-5887	430	22	)	)	PUNCT
ejpam-5887	430	23	andm	andm	PROPN
ejpam-5887	430	24	≡	≡	PROPN
ejpam-5887	430	25	r	r	PROPN
ejpam-5887	430	26	(	(	PUNCT
ejpam-5887	430	27	mod	mod	PROPN
ejpam-5887	430	28	k	k	PROPN
ejpam-5887	430	29	)	)	PUNCT
ejpam-5887	430	30	;	;	PUNCT
ejpam-5887	430	31	0	0	NUM
ejpam-5887	430	32	≤	≤	NUM
ejpam-5887	430	33	r	r	NOUN
ejpam-5887	430	34	≤	≤	NUM
ejpam-5887	430	35	k	k	NOUN
ejpam-5887	430	36	−	−	NOUN
ejpam-5887	430	37	1	1	NUM
ejpam-5887	430	38	as	as	SCONJ
ejpam-5887	430	39	follows	follow	VERB
ejpam-5887	430	40	:	:	PUNCT
ejpam-5887	430	41	f(vivi+1	f(vivi+1	X
ejpam-5887	430	42	)	)	PUNCT
ejpam-5887	430	43	=	=	SYM
ejpam-5887	430	44	0	0	NUM
ejpam-5887	430	45	;	;	PUNCT
ejpam-5887	430	46	1	1	NUM
ejpam-5887	430	47	≤	≤	NUM
ejpam-5887	431	1	i	i	PRON
ejpam-5887	431	2	≤	≤	ADJ
ejpam-5887	431	3	n−	n−	PROPN
ejpam-5887	431	4	1	1	NUM
ejpam-5887	431	5	,	,	PUNCT
ejpam-5887	431	6	f(viui	f(viui	NOUN
ejpam-5887	431	7	)	)	PUNCT
ejpam-5887	431	8	=	=	SYM
ejpam-5887	431	9	0	0	NUM
ejpam-5887	431	10	;	;	PUNCT
ejpam-5887	431	11	1	1	NUM
ejpam-5887	431	12	≤	≤	NUM
ejpam-5887	431	13	i	i	PRON
ejpam-5887	431	14	≤	≤	PROPN
ejpam-5887	431	15	n	n	CCONJ
ejpam-5887	431	16	,	,	PUNCT
ejpam-5887	431	17	we	we	PRON
ejpam-5887	431	18	have	have	VERB
ejpam-5887	431	19	the	the	DET
ejpam-5887	431	20	following	follow	VERB
ejpam-5887	431	21	two	two	NUM
ejpam-5887	431	22	cases	case	NOUN
ejpam-5887	431	23	.	.	PUNCT
ejpam-5887	432	1	case	case	NOUN
ejpam-5887	432	2	(	(	PUNCT
ejpam-5887	432	3	i	i	NOUN
ejpam-5887	432	4	):	):	PUNCT
ejpam-5887	432	5	if	if	SCONJ
ejpam-5887	432	6	n	n	PRON
ejpam-5887	432	7	≡	≡	PROPN
ejpam-5887	432	8	0	0	PUNCT
ejpam-5887	432	9	(	(	PUNCT
ejpam-5887	432	10	mod	mod	PROPN
ejpam-5887	432	11	k	k	PROPN
ejpam-5887	432	12	)	)	PUNCT
ejpam-5887	432	13	,	,	PUNCT
ejpam-5887	432	14	then	then	ADV
ejpam-5887	432	15	f(viv	f(viv	PROPN
ejpam-5887	433	1	j	j	PROPN
ejpam-5887	433	2	i	i	NOUN
ejpam-5887	433	3	)	)	PUNCT
ejpam-5887	434	1	=	=	PUNCT
ejpam-5887	434	2	0	0	NUM
ejpam-5887	434	3	;	;	PUNCT
ejpam-5887	434	4	1	1	NUM
ejpam-5887	434	5	≤	≤	NUM
ejpam-5887	434	6	i	i	PRON
ejpam-5887	434	7	≤	≤	ADJ
ejpam-5887	435	1	n	n	CCONJ
ejpam-5887	435	2	,	,	PUNCT
ejpam-5887	435	3	1	1	NUM
ejpam-5887	435	4	≤	≤	NUM
ejpam-5887	435	5	j	j	PROPN
ejpam-5887	435	6	≤	≤	PROPN
ejpam-5887	435	7	⌊mk	⌊mk	NOUN
ejpam-5887	435	8	⌋	⌋	NOUN
ejpam-5887	435	9	−	−	PROPN
ejpam-5887	435	10	1	1	NUM
ejpam-5887	435	11	,	,	PUNCT
ejpam-5887	435	12	f(viv	f(viv	PROPN
ejpam-5887	435	13	⌊m	⌊m	ADP
ejpam-5887	436	1	k	k	PROPN
ejpam-5887	436	2	⌋	⌋	PROPN
ejpam-5887	436	3	i	i	NOUN
ejpam-5887	436	4	)	)	PUNCT
ejpam-5887	437	1	=	=	PUNCT
ejpam-5887	437	2			NUM
ejpam-5887	437	3	0	0	NUM
ejpam-5887	437	4	;	;	PUNCT
ejpam-5887	437	5	1	1	NUM
ejpam-5887	437	6	≤	≤	NUM
ejpam-5887	437	7	i	i	PRON
ejpam-5887	437	8	≤	≤	NOUN
ejpam-5887	437	9	(	(	PUNCT
ejpam-5887	437	10	r+1)n	r+1)n	PROPN
ejpam-5887	437	11	k	k	PROPN
ejpam-5887	437	12	1	1	NUM
ejpam-5887	437	13	;	;	PUNCT
ejpam-5887	437	14	(	(	PUNCT
ejpam-5887	437	15	r+1)n	r+1)n	PROPN
ejpam-5887	437	16	k	k	PROPN
ejpam-5887	438	1	+	+	PROPN
ejpam-5887	438	2	1	1	X
ejpam-5887	438	3	≤	≤	NUM
ejpam-5887	438	4	i	i	PRON
ejpam-5887	438	5	≤	≤	NOUN
ejpam-5887	438	6	(	(	PUNCT
ejpam-5887	438	7	r+2)n	r+2)n	NOUN
ejpam-5887	438	8	k	k	PROPN
ejpam-5887	438	9	2	2	NUM
ejpam-5887	438	10	;	;	PUNCT
ejpam-5887	438	11	(	(	PUNCT
ejpam-5887	438	12	r+2)n	r+2)n	NOUN
ejpam-5887	438	13	k	k	PROPN
ejpam-5887	439	1	+	+	CCONJ
ejpam-5887	439	2	1	1	X
ejpam-5887	439	3	≤	≤	NUM
ejpam-5887	439	4	i	i	PRON
ejpam-5887	439	5	≤	≤	NOUN
ejpam-5887	439	6	(	(	PUNCT
ejpam-5887	439	7	r+3)n	r+3)n	PROPN
ejpam-5887	439	8	k	k	NOUN
ejpam-5887	439	9	:	:	PUNCT
ejpam-5887	439	10	:	:	PUNCT
ejpam-5887	440	1	k	k	X
ejpam-5887	440	2	−	−	NOUN
ejpam-5887	440	3	r	r	NOUN
ejpam-5887	440	4	−	−	NOUN
ejpam-5887	440	5	1	1	NUM
ejpam-5887	440	6	;	;	PUNCT
ejpam-5887	440	7	(	(	PUNCT
ejpam-5887	440	8	k−1)n	k−1)n	PROPN
ejpam-5887	440	9	k	k	PROPN
ejpam-5887	440	10	≤	≤	PROPN
ejpam-5887	440	11	i	i	PRON
ejpam-5887	440	12	≤	≤	ADJ
ejpam-5887	440	13	n	n	X
ejpam-5887	440	14	;	;	PUNCT
ejpam-5887	440	15	m	m	VERB
ejpam-5887	440	16	̸≡	̸≡	NOUN
ejpam-5887	441	1	k	k	PROPN
ejpam-5887	442	1	−	−	PROPN
ejpam-5887	442	2	1	1	NUM
ejpam-5887	442	3	(	(	PUNCT
ejpam-5887	442	4	mod	mod	PROPN
ejpam-5887	442	5	k	k	PROPN
ejpam-5887	442	6	)	)	PUNCT
ejpam-5887	442	7	,	,	PUNCT
ejpam-5887	442	8	f(viv	f(viv	PROPN
ejpam-5887	442	9	⌊m	⌊m	VERB
ejpam-5887	443	1	k	k	PROPN
ejpam-5887	443	2	⌋+j	⌋+j	PROPN
ejpam-5887	444	1	i	i	INTJ
ejpam-5887	444	2	)	)	PUNCT
ejpam-5887	445	1	=	=	PUNCT
ejpam-5887	446	1			PUNCT
ejpam-5887	446	2	q	q	NOUN
ejpam-5887	446	3	;	;	PUNCT
ejpam-5887	446	4	j	j	PROPN
ejpam-5887	446	5	≡	≡	PROPN
ejpam-5887	446	6	q	q	PROPN
ejpam-5887	447	1	(	(	PUNCT
ejpam-5887	447	2	mod	mod	NOUN
ejpam-5887	447	3	k	k	PROPN
ejpam-5887	447	4	−	−	PROPN
ejpam-5887	447	5	1	1	NUM
ejpam-5887	447	6	)	)	PUNCT
ejpam-5887	447	7	,	,	PUNCT
ejpam-5887	447	8	1	1	NUM
ejpam-5887	447	9	≤	≤	NUM
ejpam-5887	447	10	q	q	PROPN
ejpam-5887	447	11	≤	≤	NUM
ejpam-5887	447	12	k	k	NOUN
ejpam-5887	448	1	−	−	PROPN
ejpam-5887	448	2	2	2	NUM
ejpam-5887	448	3	k	k	NOUN
ejpam-5887	448	4	−	−	PROPN
ejpam-5887	448	5	1	1	NUM
ejpam-5887	448	6	;	;	PUNCT
ejpam-5887	448	7	j	j	PROPN
ejpam-5887	448	8	≡	≡	PROPN
ejpam-5887	448	9	0	0	PUNCT
ejpam-5887	448	10	(	(	PUNCT
ejpam-5887	448	11	mod	mod	NOUN
ejpam-5887	448	12	k	k	PROPN
ejpam-5887	448	13	−	−	PROPN
ejpam-5887	449	1	1	1	NUM
ejpam-5887	449	2	)	)	PUNCT
ejpam-5887	449	3	;	;	PUNCT
ejpam-5887	449	4	1	1	NUM
ejpam-5887	449	5	≤	≤	NUM
ejpam-5887	449	6	i	i	PRON
ejpam-5887	449	7	≤	≤	ADJ
ejpam-5887	449	8	n	n	CCONJ
ejpam-5887	449	9	,	,	PUNCT
ejpam-5887	449	10	1	1	NUM
ejpam-5887	449	11	≤	≤	NUM
ejpam-5887	449	12	j	j	PROPN
ejpam-5887	449	13	≤	≤	PROPN
ejpam-5887	449	14	(	(	PUNCT
ejpam-5887	449	15	k	k	PROPN
ejpam-5887	449	16	−	−	PROPN
ejpam-5887	449	17	1)⌊mk	1)⌊mk	NUM
ejpam-5887	449	18	⌋	⌋	NOUN
ejpam-5887	449	19	,	,	PUNCT
ejpam-5887	449	20	f(viv	f(viv	PROPN
ejpam-5887	449	21	k⌊m	k⌊m	PROPN
ejpam-5887	449	22	k	k	PROPN
ejpam-5887	449	23	⌋+j	⌋+j	PROPN
ejpam-5887	449	24	i	i	NOUN
ejpam-5887	449	25	)	)	PUNCT
ejpam-5887	449	26	=	=	PUNCT
ejpam-5887	450	1			PUNCT
ejpam-5887	450	2	k	k	INTJ
ejpam-5887	450	3	−	−	PROPN
ejpam-5887	450	4	q	q	NOUN
ejpam-5887	450	5	;	;	PUNCT
ejpam-5887	450	6	j	j	PROPN
ejpam-5887	450	7	≡	≡	PROPN
ejpam-5887	450	8	q	q	PROPN
ejpam-5887	451	1	(	(	PUNCT
ejpam-5887	451	2	mod	mod	NOUN
ejpam-5887	451	3	k	k	PROPN
ejpam-5887	451	4	−	−	PROPN
ejpam-5887	451	5	1	1	NUM
ejpam-5887	451	6	)	)	PUNCT
ejpam-5887	451	7	,	,	PUNCT
ejpam-5887	451	8	1	1	NUM
ejpam-5887	451	9	≤	≤	NUM
ejpam-5887	451	10	q	q	PROPN
ejpam-5887	451	11	≤	≤	NUM
ejpam-5887	451	12	k	k	NOUN
ejpam-5887	452	1	−	−	NUM
ejpam-5887	452	2	2	2	NUM
ejpam-5887	452	3	1	1	NUM
ejpam-5887	452	4	;	;	PUNCT
ejpam-5887	452	5	j	j	PROPN
ejpam-5887	452	6	≡	≡	PROPN
ejpam-5887	452	7	0	0	PUNCT
ejpam-5887	452	8	(	(	PUNCT
ejpam-5887	452	9	mod	mod	NOUN
ejpam-5887	452	10	k	k	PROPN
ejpam-5887	452	11	−	−	PROPN
ejpam-5887	453	1	1	1	NUM
ejpam-5887	453	2	)	)	PUNCT
ejpam-5887	453	3	;	;	PUNCT
ejpam-5887	453	4	1	1	NUM
ejpam-5887	453	5	≤	≤	NUM
ejpam-5887	453	6	i	i	PRON
ejpam-5887	453	7	≤	≤	NOUN
ejpam-5887	453	8	n	n	CCONJ
ejpam-5887	453	9	k	k	NOUN
ejpam-5887	453	10	,	,	PUNCT
ejpam-5887	453	11	1	1	NUM
ejpam-5887	453	12	≤	≤	NUM
ejpam-5887	453	13	j	j	PROPN
ejpam-5887	453	14	≤	≤	PROPN
ejpam-5887	453	15	r	r	NOUN
ejpam-5887	453	16	,	,	PUNCT
ejpam-5887	453	17	f(vn	f(vn	PROPN
ejpam-5887	453	18	k	k	PROPN
ejpam-5887	454	1	+	+	PROPN
ejpam-5887	454	2	iv	iv	PROPN
ejpam-5887	454	3	k⌊m	k⌊m	PROPN
ejpam-5887	454	4	k	k	PROPN
ejpam-5887	454	5	⌋+j	⌋+j	PROPN
ejpam-5887	454	6	n	n	PROPN
ejpam-5887	455	1	k	k	PROPN
ejpam-5887	455	2	+	+	PROPN
ejpam-5887	455	3	i	i	NOUN
ejpam-5887	455	4	)	)	PUNCT
ejpam-5887	456	1	=	=	PUNCT
ejpam-5887	456	2			PUNCT
ejpam-5887	456	3	q	q	NOUN
ejpam-5887	456	4	;	;	PUNCT
ejpam-5887	456	5	i	i	PRON
ejpam-5887	456	6	≡	≡	PROPN
ejpam-5887	456	7	q	q	X
ejpam-5887	457	1	(	(	PUNCT
ejpam-5887	457	2	mod	mod	PROPN
ejpam-5887	457	3	k	k	PROPN
ejpam-5887	457	4	−	−	PROPN
ejpam-5887	457	5	1	1	NUM
ejpam-5887	457	6	)	)	PUNCT
ejpam-5887	457	7	,	,	PUNCT
ejpam-5887	457	8	1	1	NUM
ejpam-5887	457	9	≤	≤	NUM
ejpam-5887	457	10	q	q	PROPN
ejpam-5887	457	11	≤	≤	NUM
ejpam-5887	457	12	k	k	NOUN
ejpam-5887	458	1	−	−	PROPN
ejpam-5887	458	2	2	2	NUM
ejpam-5887	458	3	k	k	NOUN
ejpam-5887	458	4	−	−	PROPN
ejpam-5887	458	5	1	1	NUM
ejpam-5887	458	6	;	;	PUNCT
ejpam-5887	458	7	i	i	PRON
ejpam-5887	458	8	≡	≡	PROPN
ejpam-5887	458	9	0	0	PUNCT
ejpam-5887	459	1	(	(	PUNCT
ejpam-5887	459	2	mod	mod	NOUN
ejpam-5887	459	3	k	k	PROPN
ejpam-5887	459	4	−	−	PROPN
ejpam-5887	459	5	1	1	NUM
ejpam-5887	459	6	)	)	PUNCT
ejpam-5887	459	7	;	;	PUNCT
ejpam-5887	459	8	1	1	NUM
ejpam-5887	459	9	≤	≤	NUM
ejpam-5887	459	10	i	i	PRON
ejpam-5887	459	11	≤	≤	NOUN
ejpam-5887	459	12	(	(	PUNCT
ejpam-5887	459	13	k−1)n	k−1)n	PROPN
ejpam-5887	459	14	k	k	PROPN
ejpam-5887	459	15	,	,	PUNCT
ejpam-5887	459	16	1	1	NUM
ejpam-5887	459	17	≤	≤	NUM
ejpam-5887	459	18	j	j	PROPN
ejpam-5887	459	19	≤	≤	ADJ
ejpam-5887	459	20	r	r	NOUN
ejpam-5887	459	21	,	,	PUNCT
ejpam-5887	459	22	n.	n.	NOUN
ejpam-5887	459	23	m.	m.	NOUN
ejpam-5887	459	24	noureldeen	noureldeen	NOUN
ejpam-5887	459	25	et	et	PROPN
ejpam-5887	459	26	al	al	PROPN
ejpam-5887	459	27	.	.	PUNCT
ejpam-5887	459	28	/	/	SYM
ejpam-5887	459	29	eur	eur	PROPN
ejpam-5887	459	30	.	.	PUNCT
ejpam-5887	460	1	j.	j.	PROPN
ejpam-5887	460	2	pure	pure	PROPN
ejpam-5887	460	3	appl	appl	PROPN
ejpam-5887	460	4	.	.	PROPN
ejpam-5887	460	5	math	math	PROPN
ejpam-5887	460	6	,	,	PUNCT
ejpam-5887	460	7	18	18	NUM
ejpam-5887	460	8	(	(	PUNCT
ejpam-5887	460	9	2	2	NUM
ejpam-5887	460	10	)	)	PUNCT
ejpam-5887	460	11	(	(	PUNCT
ejpam-5887	460	12	2025	2025	NUM
ejpam-5887	460	13	)	)	PUNCT
ejpam-5887	460	14	,	,	PUNCT
ejpam-5887	460	15	5887	5887	NUM
ejpam-5887	460	16	14	14	NUM
ejpam-5887	460	17	of	of	ADP
ejpam-5887	460	18	21	21	NUM
ejpam-5887	460	19	f(uiu	f(uiu	PROPN
ejpam-5887	460	20	j	j	PROPN
ejpam-5887	460	21	i	i	PROPN
ejpam-5887	460	22	)	)	PUNCT
ejpam-5887	461	1	=	=	PUNCT
ejpam-5887	462	1	f(viv	f(viv	PROPN
ejpam-5887	462	2	j	j	PROPN
ejpam-5887	462	3	i	i	PROPN
ejpam-5887	462	4	)	)	PUNCT
ejpam-5887	462	5	;	;	PUNCT
ejpam-5887	462	6	1	1	NUM
ejpam-5887	462	7	≤	≤	NUM
ejpam-5887	462	8	i	i	PRON
ejpam-5887	462	9	≤	≤	ADJ
ejpam-5887	463	1	n	n	CCONJ
ejpam-5887	463	2	,	,	PUNCT
ejpam-5887	463	3	1	1	NUM
ejpam-5887	463	4	≤	≤	NUM
ejpam-5887	463	5	j	j	PROPN
ejpam-5887	463	6	≤	≤	PROPN
ejpam-5887	463	7	m.	m.	NOUN
ejpam-5887	463	8	from	from	ADP
ejpam-5887	463	9	this	this	DET
ejpam-5887	463	10	labeling	labeling	NOUN
ejpam-5887	463	11	we	we	PRON
ejpam-5887	463	12	have	have	VERB
ejpam-5887	463	13	,	,	PUNCT
ejpam-5887	463	14	ef	ef	PROPN
ejpam-5887	463	15	(	(	PUNCT
ejpam-5887	463	16	i	i	NOUN
ejpam-5887	463	17	)	)	PUNCT
ejpam-5887	463	18	=	=	PRON
ejpam-5887	463	19	{	{	PUNCT
ejpam-5887	463	20	2n	2n	NUM
ejpam-5887	463	21	k	k	NOUN
ejpam-5887	464	1	+	+	CCONJ
ejpam-5887	465	1	2n⌊mk	2n⌊mk	NUM
ejpam-5887	465	2	⌋+	⌋+	NUM
ejpam-5887	465	3	2rn	2rn	NOUN
ejpam-5887	466	1	k	k	X
ejpam-5887	467	1	−	−	NOUN
ejpam-5887	467	2	1	1	NUM
ejpam-5887	467	3	;	;	PUNCT
ejpam-5887	467	4	i	i	PRON
ejpam-5887	467	5	=	=	SYM
ejpam-5887	467	6	0	0	NUM
ejpam-5887	467	7	2n	2n	NUM
ejpam-5887	467	8	k	k	X
ejpam-5887	468	1	+	+	CCONJ
ejpam-5887	468	2	2n⌊mk	2n⌊mk	NUM
ejpam-5887	468	3	⌋+	⌋+	NUM
ejpam-5887	469	1	2rn	2rn	PROPN
ejpam-5887	469	2	k	k	X
ejpam-5887	469	3	;	;	PUNCT
ejpam-5887	469	4	1	1	NUM
ejpam-5887	469	5	≤	≤	NUM
ejpam-5887	469	6	i	i	X
ejpam-5887	469	7	≤	≤	NOUN
ejpam-5887	470	1	k	k	PRON
ejpam-5887	470	2	−	−	PROPN
ejpam-5887	470	3	1	1	NUM
ejpam-5887	470	4	,	,	PUNCT
ejpam-5887	470	5	vf∗(i	vf∗(i	ADJ
ejpam-5887	470	6	)	)	PUNCT
ejpam-5887	470	7	=	=	SYM
ejpam-5887	471	1	2n	2n	NUM
ejpam-5887	472	1	k	k	NOUN
ejpam-5887	472	2	+	+	CCONJ
ejpam-5887	472	3	2n⌊mk	2n⌊mk	NUM
ejpam-5887	472	4	⌋+	⌋+	NUM
ejpam-5887	473	1	2rn	2rn	PROPN
ejpam-5887	473	2	k	k	X
ejpam-5887	473	3	;	;	PUNCT
ejpam-5887	473	4	0	0	NUM
ejpam-5887	473	5	≤	≤	NUM
ejpam-5887	473	6	i	i	PRON
ejpam-5887	473	7	≤	≤	NOUN
ejpam-5887	474	1	k	k	PRON
ejpam-5887	475	1	−	−	NOUN
ejpam-5887	475	2	1	1	X
ejpam-5887	475	3	.	.	PUNCT
ejpam-5887	475	4	case	case	NOUN
ejpam-5887	475	5	(	(	PUNCT
ejpam-5887	475	6	ii	ii	NOUN
ejpam-5887	475	7	):	):	PUNCT
ejpam-5887	475	8	if	if	SCONJ
ejpam-5887	475	9	n	n	PRON
ejpam-5887	475	10	≡	≡	PROPN
ejpam-5887	475	11	1	1	NUM
ejpam-5887	475	12	(	(	PUNCT
ejpam-5887	475	13	mod	mod	PROPN
ejpam-5887	475	14	k	k	PROPN
ejpam-5887	475	15	)	)	PUNCT
ejpam-5887	475	16	,	,	PUNCT
ejpam-5887	475	17	then	then	ADV
ejpam-5887	475	18	f(viv	f(viv	PROPN
ejpam-5887	475	19	j	j	PROPN
ejpam-5887	475	20	i	i	PROPN
ejpam-5887	475	21	)	)	PUNCT
ejpam-5887	475	22	;	;	PUNCT
ejpam-5887	475	23	1	1	NUM
ejpam-5887	475	24	≤	≤	NUM
ejpam-5887	475	25	i	i	PRON
ejpam-5887	475	26	≤	≤	ADJ
ejpam-5887	475	27	n−	n−	NOUN
ejpam-5887	475	28	1	1	NUM
ejpam-5887	475	29	,	,	PUNCT
ejpam-5887	475	30	1	1	NUM
ejpam-5887	475	31	≤	≤	NUM
ejpam-5887	475	32	j	j	PROPN
ejpam-5887	475	33	≤	≤	NUM
ejpam-5887	475	34	m	m	VERB
ejpam-5887	475	35	as	as	ADP
ejpam-5887	475	36	in	in	ADP
ejpam-5887	475	37	case	case	NOUN
ejpam-5887	475	38	(	(	PUNCT
ejpam-5887	475	39	i	i	NOUN
ejpam-5887	475	40	)	)	PUNCT
ejpam-5887	475	41	,	,	PUNCT
ejpam-5887	475	42	f(uiu	f(uiu	ADP
ejpam-5887	475	43	j	j	PROPN
ejpam-5887	475	44	i	i	PROPN
ejpam-5887	475	45	)	)	PUNCT
ejpam-5887	475	46	;	;	PUNCT
ejpam-5887	475	47	1	1	NUM
ejpam-5887	475	48	≤	≤	NUM
ejpam-5887	475	49	i	i	PRON
ejpam-5887	475	50	≤	≤	ADJ
ejpam-5887	475	51	n−	n−	NOUN
ejpam-5887	475	52	1	1	NUM
ejpam-5887	475	53	,	,	PUNCT
ejpam-5887	475	54	1	1	NUM
ejpam-5887	475	55	≤	≤	NUM
ejpam-5887	475	56	j	j	PROPN
ejpam-5887	475	57	≤	≤	NUM
ejpam-5887	475	58	m	m	VERB
ejpam-5887	475	59	as	as	ADP
ejpam-5887	475	60	in	in	ADP
ejpam-5887	475	61	case	case	NOUN
ejpam-5887	475	62	(	(	PUNCT
ejpam-5887	475	63	i	i	NOUN
ejpam-5887	475	64	)	)	PUNCT
ejpam-5887	475	65	,	,	PUNCT
ejpam-5887	475	66	f(vnv	f(vnv	PROPN
ejpam-5887	475	67	j	j	PROPN
ejpam-5887	475	68	n	n	CCONJ
ejpam-5887	475	69	)	)	PUNCT
ejpam-5887	475	70	=	=	SYM
ejpam-5887	475	71	0	0	NUM
ejpam-5887	475	72	;	;	PUNCT
ejpam-5887	475	73	1	1	NUM
ejpam-5887	475	74	≤	≤	NUM
ejpam-5887	475	75	j	j	PROPN
ejpam-5887	475	76	≤	≤	PROPN
ejpam-5887	475	77	⌊mk	⌊mk	NOUN
ejpam-5887	475	78	⌋	⌋	PROPN
ejpam-5887	475	79	,	,	PUNCT
ejpam-5887	475	80	f(unu	f(unu	PROPN
ejpam-5887	475	81	j	j	PROPN
ejpam-5887	475	82	n	n	CCONJ
ejpam-5887	475	83	)	)	PUNCT
ejpam-5887	475	84	=	=	SYM
ejpam-5887	475	85	0	0	NUM
ejpam-5887	475	86	;	;	PUNCT
ejpam-5887	475	87	1	1	NUM
ejpam-5887	475	88	≤	≤	NUM
ejpam-5887	475	89	j	j	PROPN
ejpam-5887	475	90	≤	≤	PROPN
ejpam-5887	475	91	⌊mk	⌊mk	NOUN
ejpam-5887	475	92	⌋	⌋	NOUN
ejpam-5887	475	93	−	−	PROPN
ejpam-5887	475	94	1	1	NUM
ejpam-5887	475	95	,	,	PUNCT
ejpam-5887	475	96	f(vnv	f(vnv	NOUN
ejpam-5887	475	97	⌊m	⌊m	ADP
ejpam-5887	476	1	k	k	PROPN
ejpam-5887	476	2	⌋+j	⌋+j	PROPN
ejpam-5887	476	3	n	n	CCONJ
ejpam-5887	476	4	)	)	PUNCT
ejpam-5887	476	5	=	=	PRON
ejpam-5887	476	6	{	{	PUNCT
ejpam-5887	476	7	q	q	NOUN
ejpam-5887	476	8	;	;	PUNCT
ejpam-5887	476	9	j	j	PROPN
ejpam-5887	476	10	≡	≡	PROPN
ejpam-5887	476	11	q	q	PROPN
ejpam-5887	477	1	(	(	PUNCT
ejpam-5887	477	2	mod	mod	NOUN
ejpam-5887	477	3	k	k	PROPN
ejpam-5887	477	4	−	−	PROPN
ejpam-5887	477	5	1	1	NUM
ejpam-5887	477	6	)	)	PUNCT
ejpam-5887	477	7	,	,	PUNCT
ejpam-5887	477	8	1	1	NUM
ejpam-5887	477	9	≤	≤	NUM
ejpam-5887	477	10	q	q	PROPN
ejpam-5887	477	11	≤	≤	NUM
ejpam-5887	477	12	k	k	NOUN
ejpam-5887	478	1	−	−	PROPN
ejpam-5887	478	2	2	2	NUM
ejpam-5887	478	3	k	k	NOUN
ejpam-5887	478	4	−	−	PROPN
ejpam-5887	478	5	1	1	NUM
ejpam-5887	478	6	;	;	PUNCT
ejpam-5887	478	7	j	j	PROPN
ejpam-5887	478	8	≡	≡	PROPN
ejpam-5887	478	9	0	0	PUNCT
ejpam-5887	478	10	(	(	PUNCT
ejpam-5887	478	11	mod	mod	NOUN
ejpam-5887	478	12	k	k	PROPN
ejpam-5887	478	13	−	−	PROPN
ejpam-5887	479	1	1	1	NUM
ejpam-5887	479	2	)	)	PUNCT
ejpam-5887	479	3	;	;	PUNCT
ejpam-5887	479	4	1	1	NUM
ejpam-5887	479	5	≤	≤	NUM
ejpam-5887	479	6	j	j	PROPN
ejpam-5887	479	7	≤	≤	PROPN
ejpam-5887	479	8	m−	m−	PROPN
ejpam-5887	479	9	⌊mk	⌊mk	PROPN
ejpam-5887	479	10	⌋	⌋	PROPN
ejpam-5887	479	11	,	,	PUNCT
ejpam-5887	479	12	f(unu	f(unu	PROPN
ejpam-5887	479	13	⌊m	⌊m	PROPN
ejpam-5887	479	14	k	k	PROPN
ejpam-5887	479	15	⌋	⌋	PROPN
ejpam-5887	479	16	n	n	PROPN
ejpam-5887	479	17	)	)	PUNCT
ejpam-5887	479	18	=	=	PUNCT
ejpam-5887	479	19			PROPN
ejpam-5887	479	20	0	0	NUM
ejpam-5887	479	21	;	;	PUNCT
ejpam-5887	479	22	m	m	VERB
ejpam-5887	479	23	≡	≡	PROPN
ejpam-5887	479	24	1	1	NUM
ejpam-5887	479	25	,	,	PUNCT
ejpam-5887	479	26	2	2	NUM
ejpam-5887	479	27	(	(	PUNCT
ejpam-5887	479	28	mod	mod	NOUN
ejpam-5887	479	29	3	3	NUM
ejpam-5887	479	30	)	)	PUNCT
ejpam-5887	479	31	,	,	PUNCT
ejpam-5887	480	1	k	k	X
ejpam-5887	480	2	=	=	SYM
ejpam-5887	480	3	3	3	NUM
ejpam-5887	480	4	;	;	PUNCT
ejpam-5887	480	5	m	m	VERB
ejpam-5887	480	6	̸≡	̸≡	NOUN
ejpam-5887	480	7	0	0	NUM
ejpam-5887	480	8	,	,	PUNCT
ejpam-5887	480	9	1	1	NUM
ejpam-5887	480	10	(	(	PUNCT
ejpam-5887	480	11	mod	mod	NOUN
ejpam-5887	480	12	k	k	PROPN
ejpam-5887	480	13	)	)	PUNCT
ejpam-5887	480	14	,	,	PUNCT
ejpam-5887	481	1	k	k	X
ejpam-5887	481	2	>	>	X
ejpam-5887	481	3	3	3	NUM
ejpam-5887	481	4	1	1	NUM
ejpam-5887	481	5	;	;	PUNCT
ejpam-5887	481	6	m	m	VERB
ejpam-5887	481	7	≡	≡	PROPN
ejpam-5887	481	8	0	0	PUNCT
ejpam-5887	482	1	(	(	PUNCT
ejpam-5887	482	2	mod	mod	PROPN
ejpam-5887	482	3	k	k	PROPN
ejpam-5887	482	4	)	)	PUNCT
ejpam-5887	482	5	,	,	PUNCT
ejpam-5887	483	1	k	k	PROPN
ejpam-5887	483	2	≥	≥	NUM
ejpam-5887	483	3	3	3	NUM
ejpam-5887	483	4	k	k	NOUN
ejpam-5887	483	5	−	−	PROPN
ejpam-5887	483	6	2	2	NUM
ejpam-5887	483	7	;	;	PUNCT
ejpam-5887	483	8	k	k	PROPN
ejpam-5887	483	9	=	=	SYM
ejpam-5887	483	10	2	2	NUM
ejpam-5887	483	11	;	;	PUNCT
ejpam-5887	483	12	m	m	VERB
ejpam-5887	483	13	≡	≡	PROPN
ejpam-5887	483	14	1	1	NUM
ejpam-5887	483	15	(	(	PUNCT
ejpam-5887	483	16	mod	mod	NOUN
ejpam-5887	483	17	k	k	PROPN
ejpam-5887	483	18	)	)	PUNCT
ejpam-5887	483	19	,	,	PUNCT
ejpam-5887	484	1	k	k	X
ejpam-5887	484	2	>	>	X
ejpam-5887	484	3	3	3	NUM
ejpam-5887	484	4	,	,	PUNCT
ejpam-5887	484	5	f(unu	f(unu	PROPN
ejpam-5887	484	6	⌊m	⌊m	PROPN
ejpam-5887	484	7	k	k	PROPN
ejpam-5887	484	8	⌋+j	⌋+j	PROPN
ejpam-5887	484	9	n	n	CCONJ
ejpam-5887	484	10	)	)	PUNCT
ejpam-5887	484	11	=	=	PRON
ejpam-5887	484	12	{	{	PUNCT
ejpam-5887	485	1	k	k	NOUN
ejpam-5887	485	2	−	−	PROPN
ejpam-5887	485	3	q	q	NOUN
ejpam-5887	485	4	;	;	PUNCT
ejpam-5887	485	5	j	j	PROPN
ejpam-5887	485	6	≡	≡	PROPN
ejpam-5887	485	7	q	q	PROPN
ejpam-5887	485	8	(	(	PUNCT
ejpam-5887	485	9	mod	mod	NOUN
ejpam-5887	485	10	k	k	PROPN
ejpam-5887	485	11	−	−	PROPN
ejpam-5887	485	12	1	1	NUM
ejpam-5887	485	13	)	)	PUNCT
ejpam-5887	485	14	,	,	PUNCT
ejpam-5887	485	15	1	1	NUM
ejpam-5887	485	16	≤	≤	NUM
ejpam-5887	485	17	q	q	PROPN
ejpam-5887	485	18	≤	≤	NUM
ejpam-5887	485	19	k	k	NOUN
ejpam-5887	486	1	−	−	NUM
ejpam-5887	487	1	2	2	NUM
ejpam-5887	487	2	1	1	NUM
ejpam-5887	487	3	;	;	PUNCT
ejpam-5887	487	4	j	j	PROPN
ejpam-5887	487	5	≡	≡	PROPN
ejpam-5887	487	6	0	0	PUNCT
ejpam-5887	488	1	(	(	PUNCT
ejpam-5887	488	2	mod	mod	NOUN
ejpam-5887	488	3	k	k	PROPN
ejpam-5887	488	4	−	−	PROPN
ejpam-5887	488	5	1	1	NUM
ejpam-5887	488	6	)	)	PUNCT
ejpam-5887	488	7	;	;	PUNCT
ejpam-5887	488	8	1	1	NUM
ejpam-5887	488	9	≤	≤	NUM
ejpam-5887	488	10	j	j	PROPN
ejpam-5887	488	11	≤	≤	PROPN
ejpam-5887	488	12	m−	m−	PROPN
ejpam-5887	488	13	⌊mk	⌊mk	PROPN
ejpam-5887	488	14	⌋.	⌋.	ADV
ejpam-5887	488	15	from	from	ADP
ejpam-5887	488	16	this	this	DET
ejpam-5887	488	17	labeling	labeling	NOUN
ejpam-5887	488	18	we	we	PRON
ejpam-5887	488	19	have	have	VERB
ejpam-5887	488	20	,	,	PUNCT
ejpam-5887	488	21	ef	ef	PROPN
ejpam-5887	488	22	(	(	PUNCT
ejpam-5887	488	23	i	i	NOUN
ejpam-5887	488	24	)	)	PUNCT
ejpam-5887	488	25	=	=	PRON
ejpam-5887	488	26	{	{	PUNCT
ejpam-5887	488	27	2⌊nk	2⌊nk	NUM
ejpam-5887	488	28	⌋+	⌋+	NUM
ejpam-5887	489	1	2mn	2mn	PROPN
ejpam-5887	489	2	k	k	X
ejpam-5887	489	3	;	;	PUNCT
ejpam-5887	489	4	i	i	PRON
ejpam-5887	489	5	=	=	NOUN
ejpam-5887	489	6	0	0	NUM
ejpam-5887	489	7	;	;	PUNCT
ejpam-5887	489	8	2	2	NUM
ejpam-5887	489	9	≤	≤	NUM
ejpam-5887	489	10	i	i	X
ejpam-5887	489	11	≤	≤	NOUN
ejpam-5887	490	1	k	k	PRON
ejpam-5887	491	1	−	−	NUM
ejpam-5887	491	2	1	1	NUM
ejpam-5887	491	3	2⌊nk	2⌊nk	NUM
ejpam-5887	491	4	⌋+	⌋+	PUNCT
ejpam-5887	492	1	2mn	2mn	PROPN
ejpam-5887	492	2	k	k	X
ejpam-5887	493	1	+	+	CCONJ
ejpam-5887	493	2	1	1	NUM
ejpam-5887	493	3	;	;	PUNCT
ejpam-5887	493	4	i	i	PRON
ejpam-5887	493	5	=	=	NOUN
ejpam-5887	493	6	1	1	NUM
ejpam-5887	493	7	;	;	PUNCT
ejpam-5887	493	8	m	m	VERB
ejpam-5887	493	9	≡	≡	PROPN
ejpam-5887	493	10	0	0	PUNCT
ejpam-5887	494	1	(	(	PUNCT
ejpam-5887	494	2	mod	mod	PROPN
ejpam-5887	494	3	k	k	PROPN
ejpam-5887	494	4	)	)	PUNCT
ejpam-5887	494	5	,	,	PUNCT
ejpam-5887	494	6	ef	ef	PROPN
ejpam-5887	494	7	(	(	PUNCT
ejpam-5887	494	8	i	i	NOUN
ejpam-5887	494	9	)	)	PUNCT
ejpam-5887	494	10	=	=	PUNCT
ejpam-5887	495	1			PROPN
ejpam-5887	495	2	2⌊nk	2⌊nk	NUM
ejpam-5887	495	3	⌋+	⌋+	NUM
ejpam-5887	495	4	2n⌊mk	2n⌊mk	NUM
ejpam-5887	495	5	⌋+	⌋+	NUM
ejpam-5887	495	6	2⌊nk	2⌊nk	NUM
ejpam-5887	495	7	⌋	⌋	NOUN
ejpam-5887	495	8	;	;	PUNCT
ejpam-5887	496	1	i	i	PROPN
ejpam-5887	496	2	=	=	NOUN
ejpam-5887	496	3	0	0	NUM
ejpam-5887	496	4	,	,	PUNCT
ejpam-5887	496	5	k	k	X
ejpam-5887	496	6	>	>	X
ejpam-5887	496	7	3	3	NUM
ejpam-5887	496	8	;	;	PUNCT
ejpam-5887	496	9	2	2	NUM
ejpam-5887	496	10	≤	≤	NUM
ejpam-5887	496	11	i	i	X
ejpam-5887	496	12	≤	≤	NOUN
ejpam-5887	497	1	k	k	PRON
ejpam-5887	498	1	−	−	NUM
ejpam-5887	498	2	3	3	NUM
ejpam-5887	498	3	2⌊nk	2⌊nk	NUM
ejpam-5887	498	4	⌋+	⌋+	NUM
ejpam-5887	498	5	2n⌊mk	2n⌊mk	NUM
ejpam-5887	498	6	⌋+	⌋+	NUM
ejpam-5887	498	7	2⌊nk	2⌊nk	NUM
ejpam-5887	498	8	⌋+	⌋+	ADJ
ejpam-5887	498	9	1	1	NUM
ejpam-5887	498	10	;	;	PUNCT
ejpam-5887	498	11	i	i	PRON
ejpam-5887	498	12	=	=	NOUN
ejpam-5887	498	13	1	1	NUM
ejpam-5887	498	14	,	,	PUNCT
ejpam-5887	498	15	k	k	PROPN
ejpam-5887	499	1	−	−	PROPN
ejpam-5887	499	2	1	1	NUM
ejpam-5887	499	3	,	,	PUNCT
ejpam-5887	499	4	k	k	PROPN
ejpam-5887	499	5	−	−	PROPN
ejpam-5887	499	6	2	2	NUM
ejpam-5887	499	7	;	;	PUNCT
ejpam-5887	499	8	i	i	PROPN
ejpam-5887	499	9	=	=	NOUN
ejpam-5887	499	10	0	0	NUM
ejpam-5887	499	11	,	,	PUNCT
ejpam-5887	499	12	k	k	PROPN
ejpam-5887	499	13	=	=	SYM
ejpam-5887	499	14	3	3	NUM
ejpam-5887	499	15	;	;	PUNCT
ejpam-5887	499	16	m	m	VERB
ejpam-5887	499	17	≡	≡	PROPN
ejpam-5887	499	18	1	1	NUM
ejpam-5887	499	19	(	(	PUNCT
ejpam-5887	499	20	mod	mod	NOUN
ejpam-5887	499	21	k	k	PROPN
ejpam-5887	499	22	)	)	PUNCT
ejpam-5887	499	23	,	,	PUNCT
ejpam-5887	499	24	k	k	PROPN
ejpam-5887	499	25	≥	≥	NUM
ejpam-5887	499	26	3	3	NUM
ejpam-5887	499	27	,	,	PUNCT
ejpam-5887	499	28	ef	ef	PROPN
ejpam-5887	499	29	(	(	PUNCT
ejpam-5887	499	30	i	i	NOUN
ejpam-5887	499	31	)	)	PUNCT
ejpam-5887	499	32	=	=	PRON
ejpam-5887	499	33	{	{	PUNCT
ejpam-5887	499	34	2⌊nk	2⌊nk	NUM
ejpam-5887	499	35	⌋+	⌋+	NUM
ejpam-5887	499	36	2n⌊mk	2n⌊mk	NUM
ejpam-5887	499	37	⌋+	⌋+	NUM
ejpam-5887	499	38	2(k	2(k	NUM
ejpam-5887	499	39	−	−	PROPN
ejpam-5887	499	40	1)⌊nk	1)⌊nk	NUM
ejpam-5887	499	41	⌋+	⌋+	NUM
ejpam-5887	499	42	1	1	NUM
ejpam-5887	499	43	;	;	PUNCT
ejpam-5887	499	44	i	i	PRON
ejpam-5887	499	45	=	=	NOUN
ejpam-5887	499	46	0	0	NUM
ejpam-5887	499	47	2⌊nk	2⌊nk	NUM
ejpam-5887	499	48	⌋+	⌋+	NUM
ejpam-5887	499	49	2n⌊mk	2n⌊mk	NUM
ejpam-5887	499	50	⌋+	⌋+	NUM
ejpam-5887	499	51	2(k	2(k	NUM
ejpam-5887	499	52	−	−	PROPN
ejpam-5887	499	53	1)⌊nk	1)⌊nk	NUM
ejpam-5887	499	54	⌋+	⌋+	NUM
ejpam-5887	499	55	2	2	NUM
ejpam-5887	499	56	;	;	PUNCT
ejpam-5887	499	57	1	1	NUM
ejpam-5887	499	58	≤	≤	NUM
ejpam-5887	499	59	i	i	X
ejpam-5887	499	60	≤	≤	NOUN
ejpam-5887	500	1	k	k	PRON
ejpam-5887	501	1	−	−	PROPN
ejpam-5887	501	2	1	1	NUM
ejpam-5887	501	3	;	;	PUNCT
ejpam-5887	501	4	m	m	PROPN
ejpam-5887	502	1	≡	≡	PROPN
ejpam-5887	502	2	k	k	PROPN
ejpam-5887	503	1	−	−	PROPN
ejpam-5887	503	2	1	1	NUM
ejpam-5887	503	3	(	(	PUNCT
ejpam-5887	503	4	mod	mod	PROPN
ejpam-5887	503	5	k	k	PROPN
ejpam-5887	503	6	)	)	PUNCT
ejpam-5887	503	7	,	,	PUNCT
ejpam-5887	503	8	ef	ef	PROPN
ejpam-5887	503	9	(	(	PUNCT
ejpam-5887	503	10	i	i	NOUN
ejpam-5887	503	11	)	)	PUNCT
ejpam-5887	503	12	=	=	PUNCT
ejpam-5887	503	13			PROPN
ejpam-5887	503	14	2⌊nk	2⌊nk	NUM
ejpam-5887	503	15	⌋+	⌋+	NUM
ejpam-5887	503	16	2n⌊mk	2n⌊mk	NUM
ejpam-5887	503	17	⌋+	⌋+	NUM
ejpam-5887	503	18	2r⌊nk	2r⌊nk	NUM
ejpam-5887	503	19	⌋+	⌋+	NUM
ejpam-5887	503	20	1	1	NUM
ejpam-5887	503	21	;	;	PUNCT
ejpam-5887	503	22	i	i	PRON
ejpam-5887	503	23	=	=	NOUN
ejpam-5887	503	24	0	0	NUM
ejpam-5887	503	25	,	,	PUNCT
ejpam-5887	503	26	1	1	NUM
ejpam-5887	503	27	;	;	PUNCT
ejpam-5887	503	28	2	2	NUM
ejpam-5887	503	29	≤	≤	NUM
ejpam-5887	503	30	i	i	PRON
ejpam-5887	503	31	≤	≤	NOUN
ejpam-5887	504	1	r	r	NOUN
ejpam-5887	504	2	<	<	X
ejpam-5887	504	3	k	k	X
ejpam-5887	504	4	−	−	PROPN
ejpam-5887	504	5	i	i	PRON
ejpam-5887	504	6	;	;	PUNCT
ejpam-5887	504	7	k	k	PROPN
ejpam-5887	504	8	−	−	PROPN
ejpam-5887	505	1	i	i	PRON
ejpam-5887	505	2	≤	≤	NOUN
ejpam-5887	506	1	r	r	NOUN
ejpam-5887	506	2	<	<	X
ejpam-5887	506	3	i	i	PRON
ejpam-5887	506	4	2⌊nk	2⌊nk	NUM
ejpam-5887	506	5	⌋+	⌋+	NUM
ejpam-5887	506	6	2n⌊mk	2n⌊mk	NUM
ejpam-5887	506	7	⌋+	⌋+	NUM
ejpam-5887	506	8	2r⌊nk	2r⌊nk	NUM
ejpam-5887	506	9	⌋+	⌋+	ADJ
ejpam-5887	506	10	2	2	NUM
ejpam-5887	506	11	;	;	PUNCT
ejpam-5887	506	12	i	i	PRON
ejpam-5887	506	13	≤	≤	NOUN
ejpam-5887	506	14	r	r	NOUN
ejpam-5887	506	15	,	,	PUNCT
ejpam-5887	506	16	k	k	PROPN
ejpam-5887	507	1	−	−	PROPN
ejpam-5887	507	2	i	i	PRON
ejpam-5887	507	3	≤	≤	ADJ
ejpam-5887	507	4	r	r	NOUN
ejpam-5887	507	5	;	;	PUNCT
ejpam-5887	507	6	r	r	NOUN
ejpam-5887	507	7	̸=	̸=	PROPN
ejpam-5887	507	8	0	0	NUM
ejpam-5887	507	9	,	,	PUNCT
ejpam-5887	507	10	1	1	NUM
ejpam-5887	507	11	,	,	PUNCT
ejpam-5887	507	12	k	k	PROPN
ejpam-5887	507	13	−	−	PROPN
ejpam-5887	507	14	1	1	NUM
ejpam-5887	507	15	,	,	PUNCT
ejpam-5887	507	16	k	k	X
ejpam-5887	507	17	≥	≥	NUM
ejpam-5887	507	18	3	3	NUM
ejpam-5887	507	19	,	,	PUNCT
ejpam-5887	507	20	vf∗(i	vf∗(i	ADJ
ejpam-5887	507	21	)	)	PUNCT
ejpam-5887	507	22	=	=	SYM
ejpam-5887	507	23	{	{	PUNCT
ejpam-5887	507	24	ef	ef	X
ejpam-5887	507	25	(	(	PUNCT
ejpam-5887	507	26	i	i	NOUN
ejpam-5887	507	27	)	)	PUNCT
ejpam-5887	507	28	+	+	CCONJ
ejpam-5887	507	29	1	1	NUM
ejpam-5887	507	30	;	;	PUNCT
ejpam-5887	507	31	i	i	PRON
ejpam-5887	507	32	=	=	SYM
ejpam-5887	507	33	0	0	NUM
ejpam-5887	507	34	ef	ef	PROPN
ejpam-5887	507	35	(	(	PUNCT
ejpam-5887	507	36	i	i	PROPN
ejpam-5887	507	37	)	)	PUNCT
ejpam-5887	507	38	;	;	PUNCT
ejpam-5887	507	39	1	1	NUM
ejpam-5887	507	40	≤	≤	NUM
ejpam-5887	507	41	i	i	X
ejpam-5887	507	42	≤	≤	NOUN
ejpam-5887	508	1	k	k	PRON
ejpam-5887	509	1	−	−	NOUN
ejpam-5887	509	2	1	1	X
ejpam-5887	509	3	.	.	PUNCT
ejpam-5887	510	1	clearly	clearly	ADV
ejpam-5887	510	2	,	,	PUNCT
ejpam-5887	510	3	|ef	|ef	PROPN
ejpam-5887	510	4	(	(	PUNCT
ejpam-5887	510	5	i	i	NOUN
ejpam-5887	510	6	)	)	PUNCT
ejpam-5887	510	7	−	−	PROPN
ejpam-5887	510	8	ef	ef	PROPN
ejpam-5887	510	9	(	(	PUNCT
ejpam-5887	510	10	j)|	j)|	PROPN
ejpam-5887	510	11	≤	≤	NUM
ejpam-5887	510	12	1	1	NUM
ejpam-5887	510	13	and	and	CCONJ
ejpam-5887	510	14	|vf∗(i	|vf∗(i	NUM
ejpam-5887	510	15	)	)	PUNCT
ejpam-5887	510	16	−	−	NOUN
ejpam-5887	510	17	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	510	18	≤	≤	NOUN
ejpam-5887	510	19	1	1	NUM
ejpam-5887	510	20	for	for	ADP
ejpam-5887	510	21	i	i	PRON
ejpam-5887	510	22	,	,	PUNCT
ejpam-5887	510	23	j	j	PROPN
ejpam-5887	510	24	∈	∈	PROPN
ejpam-5887	510	25	{	{	PUNCT
ejpam-5887	510	26	0	0	NUM
ejpam-5887	510	27	,	,	PUNCT
ejpam-5887	510	28	1	1	NUM
ejpam-5887	510	29	,	,	PUNCT
ejpam-5887	510	30	...	...	PUNCT
ejpam-5887	510	31	,	,	PUNCT
ejpam-5887	510	32	k	k	PROPN
ejpam-5887	510	33	−	−	PROPN
ejpam-5887	510	34	1	1	NUM
ejpam-5887	510	35	}	}	PUNCT
ejpam-5887	510	36	.	.	PUNCT
ejpam-5887	511	1	hence	hence	ADV
ejpam-5887	511	2	,	,	PUNCT
ejpam-5887	511	3	p	p	X
ejpam-5887	511	4	(	(	PUNCT
ejpam-5887	511	5	n.bv	n.bv	NOUN
ejpam-5887	511	6	m	m	PROPN
ejpam-5887	511	7	,	,	PUNCT
ejpam-5887	511	8	m	m	VERB
ejpam-5887	511	9	)	)	PUNCT
ejpam-5887	511	10	is	be	AUX
ejpam-5887	511	11	an	an	DET
ejpam-5887	511	12	edge	edge	NOUN
ejpam-5887	511	13	k	k	NOUN
ejpam-5887	511	14	-	-	PUNCT
ejpam-5887	511	15	product	product	NOUN
ejpam-5887	511	16	cordial	cordial	ADJ
ejpam-5887	511	17	graph	graph	NOUN
ejpam-5887	511	18	if	if	SCONJ
ejpam-5887	511	19	n	n	PRON
ejpam-5887	511	20	≡	≡	PROPN
ejpam-5887	511	21	0	0	NUM
ejpam-5887	511	22	,	,	PUNCT
ejpam-5887	511	23	1	1	NUM
ejpam-5887	511	24	(	(	PUNCT
ejpam-5887	511	25	mod	mod	NOUN
ejpam-5887	511	26	k	k	PROPN
ejpam-5887	511	27	)	)	PUNCT
ejpam-5887	511	28	.	.	PUNCT
ejpam-5887	512	1	example	example	NOUN
ejpam-5887	513	1	5	5	NUM
ejpam-5887	513	2	.	.	PUNCT
ejpam-5887	513	3	an	an	DET
ejpam-5887	513	4	edge	edge	NOUN
ejpam-5887	513	5	4	4	NUM
ejpam-5887	513	6	-	-	PUNCT
ejpam-5887	513	7	product	product	NOUN
ejpam-5887	513	8	cordial	cordial	ADJ
ejpam-5887	513	9	labeling	labeling	NOUN
ejpam-5887	513	10	of	of	ADP
ejpam-5887	513	11	p	p	NOUN
ejpam-5887	513	12	(	(	PUNCT
ejpam-5887	513	13	4.bv	4.bv	NOUN
ejpam-5887	513	14	5,5	5,5	NUM
ejpam-5887	513	15	)	)	PUNCT
ejpam-5887	513	16	is	be	AUX
ejpam-5887	513	17	shown	show	VERB
ejpam-5887	513	18	in	in	ADP
ejpam-5887	513	19	figure	figure	NOUN
ejpam-5887	513	20	5	5	NUM
ejpam-5887	513	21	.	.	PUNCT
ejpam-5887	514	1	n.	n.	PROPN
ejpam-5887	514	2	m.	m.	PROPN
ejpam-5887	515	1	noureldeen	noureldeen	INTJ
ejpam-5887	515	2	et	et	PROPN
ejpam-5887	515	3	al	al	PROPN
ejpam-5887	515	4	.	.	PUNCT
ejpam-5887	515	5	/	/	SYM
ejpam-5887	515	6	eur	eur	PROPN
ejpam-5887	515	7	.	.	PUNCT
ejpam-5887	516	1	j.	j.	PROPN
ejpam-5887	516	2	pure	pure	PROPN
ejpam-5887	516	3	appl	appl	PROPN
ejpam-5887	516	4	.	.	PROPN
ejpam-5887	516	5	math	math	PROPN
ejpam-5887	516	6	,	,	PUNCT
ejpam-5887	516	7	18	18	NUM
ejpam-5887	516	8	(	(	PUNCT
ejpam-5887	516	9	2	2	NUM
ejpam-5887	516	10	)	)	PUNCT
ejpam-5887	516	11	(	(	PUNCT
ejpam-5887	516	12	2025	2025	NUM
ejpam-5887	516	13	)	)	PUNCT
ejpam-5887	516	14	,	,	PUNCT
ejpam-5887	516	15	5887	5887	NUM
ejpam-5887	516	16	15	15	NUM
ejpam-5887	516	17	of	of	ADP
ejpam-5887	516	18	21	21	NUM
ejpam-5887	516	19	0	0	NUM
ejpam-5887	516	20	0	0	NUM
ejpam-5887	516	21	0	0	NUM
ejpam-5887	516	22	0	0	NUM
ejpam-5887	516	23	0	0	NUM
ejpam-5887	516	24	0	0	NUM
ejpam-5887	516	25	0	0	NUM
ejpam-5887	516	26	0	0	NUM
ejpam-5887	516	27	0	0	NUM
ejpam-5887	516	28	0	0	NUM
ejpam-5887	516	29	0	0	NUM
ejpam-5887	516	30	0	0	NUM
ejpam-5887	516	31	0	0	NUM
ejpam-5887	516	32	0	0	NUM
ejpam-5887	516	33	0	0	NUM
ejpam-5887	516	34	0	0	NUM
ejpam-5887	516	35	0	0	NUM
ejpam-5887	516	36	0	0	NUM
ejpam-5887	516	37	0	0	NUM
ejpam-5887	516	38	0	0	NUM
ejpam-5887	516	39	0	0	NUM
ejpam-5887	516	40	0	0	NUM
ejpam-5887	516	41	0	0	NUM
ejpam-5887	516	42	1	1	NUM
ejpam-5887	516	43	1	1	NUM
ejpam-5887	516	44	1	1	NUM
ejpam-5887	516	45	1	1	NUM
ejpam-5887	516	46	1	1	NUM
ejpam-5887	516	47	1	1	NUM
ejpam-5887	516	48	1	1	NUM
ejpam-5887	516	49	1	1	NUM
ejpam-5887	516	50	1	1	NUM
ejpam-5887	516	51	1	1	NUM
ejpam-5887	516	52	1	1	NUM
ejpam-5887	516	53	1	1	NUM
ejpam-5887	516	54	1	1	NUM
ejpam-5887	516	55	1	1	NUM
ejpam-5887	516	56	1	1	NUM
ejpam-5887	516	57	1	1	NUM
ejpam-5887	516	58	1	1	NUM
ejpam-5887	516	59	1	1	NUM
ejpam-5887	516	60	1	1	NUM
ejpam-5887	516	61	1	1	NUM
ejpam-5887	516	62	1	1	NUM
ejpam-5887	516	63	1	1	NUM
ejpam-5887	516	64	1	1	NUM
ejpam-5887	516	65	1	1	NUM
ejpam-5887	516	66	2	2	NUM
ejpam-5887	516	67	2	2	NUM
ejpam-5887	516	68	2	2	NUM
ejpam-5887	516	69	2	2	NUM
ejpam-5887	516	70	2	2	NUM
ejpam-5887	516	71	2	2	NUM
ejpam-5887	516	72	2	2	NUM
ejpam-5887	516	73	2	2	NUM
ejpam-5887	516	74	2	2	NUM
ejpam-5887	516	75	2	2	NUM
ejpam-5887	516	76	2	2	NUM
ejpam-5887	516	77	2	2	NUM
ejpam-5887	516	78	2	2	NUM
ejpam-5887	516	79	2	2	NUM
ejpam-5887	516	80	2	2	NUM
ejpam-5887	516	81	2	2	NUM
ejpam-5887	516	82	2	2	NUM
ejpam-5887	516	83	2	2	NUM
ejpam-5887	516	84	2	2	NUM
ejpam-5887	516	85	2	2	NUM
ejpam-5887	516	86	2	2	NUM
ejpam-5887	516	87	2	2	NUM
ejpam-5887	516	88	2	2	NUM
ejpam-5887	516	89	2	2	NUM
ejpam-5887	516	90	3	3	NUM
ejpam-5887	516	91	3	3	NUM
ejpam-5887	516	92	3	3	NUM
ejpam-5887	516	93	3	3	NUM
ejpam-5887	516	94	3	3	NUM
ejpam-5887	516	95	3	3	NUM
ejpam-5887	516	96	3	3	NUM
ejpam-5887	516	97	3	3	NUM
ejpam-5887	516	98	3	3	NUM
ejpam-5887	516	99	3	3	NUM
ejpam-5887	516	100	3	3	NUM
ejpam-5887	516	101	3	3	NUM
ejpam-5887	516	102	3	3	NUM
ejpam-5887	516	103	3	3	NUM
ejpam-5887	516	104	3	3	NUM
ejpam-5887	516	105	3	3	NUM
ejpam-5887	516	106	3	3	NUM
ejpam-5887	516	107	3	3	NUM
ejpam-5887	516	108	3	3	NUM
ejpam-5887	516	109	3	3	NUM
ejpam-5887	516	110	3	3	NUM
ejpam-5887	516	111	3	3	NUM
ejpam-5887	516	112	3	3	NUM
ejpam-5887	516	113	3	3	NUM
ejpam-5887	516	114	figure	figure	NOUN
ejpam-5887	516	115	5	5	NUM
ejpam-5887	516	116	:	:	PUNCT
ejpam-5887	516	117	edge	edge	VERB
ejpam-5887	516	118	4	4	NUM
ejpam-5887	516	119	-	-	PUNCT
ejpam-5887	516	120	product	product	NOUN
ejpam-5887	516	121	cordial	cordial	ADJ
ejpam-5887	516	122	labeling	labeling	NOUN
ejpam-5887	516	123	of	of	ADP
ejpam-5887	516	124	p	p	NOUN
ejpam-5887	516	125	(	(	PUNCT
ejpam-5887	516	126	4.bv	4.bv	NOUN
ejpam-5887	516	127	5,5	5,5	NUM
ejpam-5887	516	128	)	)	PUNCT
ejpam-5887	516	129	theorem	theorem	VERB
ejpam-5887	516	130	18	18	NUM
ejpam-5887	516	131	.	.	PUNCT
ejpam-5887	517	1	the	the	DET
ejpam-5887	517	2	path	path	PROPN
ejpam-5887	517	3	union	union	PROPN
ejpam-5887	517	4	of	of	ADP
ejpam-5887	517	5	bistar	bistar	PROPN
ejpam-5887	517	6	graph	graph	NOUN
ejpam-5887	517	7	p	p	PROPN
ejpam-5887	517	8	(	(	PUNCT
ejpam-5887	517	9	n.bv	n.bv	NOUN
ejpam-5887	517	10	m	m	PROPN
ejpam-5887	517	11	,	,	PUNCT
ejpam-5887	517	12	m	m	PROPN
ejpam-5887	517	13	)	)	PUNCT
ejpam-5887	517	14	,	,	PUNCT
ejpam-5887	517	15	where	where	SCONJ
ejpam-5887	517	16	v	v	NOUN
ejpam-5887	517	17	is	be	AUX
ejpam-5887	517	18	a	a	DET
ejpam-5887	517	19	root	root	NOUN
ejpam-5887	517	20	vertex	vertex	NOUN
ejpam-5887	517	21	of	of	ADP
ejpam-5887	517	22	bm	bm	PROPN
ejpam-5887	517	23	,	,	PUNCT
ejpam-5887	517	24	m	m	PROPN
ejpam-5887	517	25	admits	admit	VERB
ejpam-5887	517	26	an	an	DET
ejpam-5887	517	27	edge	edge	NOUN
ejpam-5887	517	28	k	k	NOUN
ejpam-5887	517	29	-	-	PUNCT
ejpam-5887	517	30	product	product	NOUN
ejpam-5887	517	31	cordial	cordial	ADJ
ejpam-5887	517	32	labeling	labeling	NOUN
ejpam-5887	517	33	if	if	SCONJ
ejpam-5887	517	34	n	n	PRON
ejpam-5887	517	35	≡	≡	PROPN
ejpam-5887	517	36	k−	k−	PROPN
ejpam-5887	517	37	1	1	NUM
ejpam-5887	517	38	(	(	PUNCT
ejpam-5887	517	39	mod	mod	NOUN
ejpam-5887	517	40	k	k	PROPN
ejpam-5887	517	41	)	)	PUNCT
ejpam-5887	517	42	and	and	CCONJ
ejpam-5887	517	43	m	m	PROPN
ejpam-5887	517	44	≡	≡	PROPN
ejpam-5887	517	45	0	0	NUM
ejpam-5887	517	46	,	,	PUNCT
ejpam-5887	517	47	k−	k−	NOUN
ejpam-5887	517	48	1	1	NUM
ejpam-5887	517	49	(	(	PUNCT
ejpam-5887	517	50	mod	mod	PROPN
ejpam-5887	517	51	k	k	PROPN
ejpam-5887	517	52	)	)	PUNCT
ejpam-5887	517	53	.	.	PUNCT
ejpam-5887	518	1	proof	proof	NOUN
ejpam-5887	518	2	.	.	PUNCT
ejpam-5887	519	1	let	let	VERB
ejpam-5887	519	2	the	the	DET
ejpam-5887	519	3	vertex	vertex	NOUN
ejpam-5887	519	4	and	and	CCONJ
ejpam-5887	519	5	edge	edge	NOUN
ejpam-5887	519	6	set	set	NOUN
ejpam-5887	519	7	of	of	ADP
ejpam-5887	519	8	p	p	PROPN
ejpam-5887	519	9	(	(	PUNCT
ejpam-5887	519	10	n.bv	n.bv	NOUN
ejpam-5887	519	11	m	m	PROPN
ejpam-5887	519	12	,	,	PUNCT
ejpam-5887	519	13	m	m	VERB
ejpam-5887	519	14	)	)	PUNCT
ejpam-5887	519	15	be	be	VERB
ejpam-5887	519	16	v	v	PRON
ejpam-5887	519	17	(	(	PUNCT
ejpam-5887	519	18	p	p	X
ejpam-5887	519	19	(	(	PUNCT
ejpam-5887	519	20	n.bv	n.bv	NOUN
ejpam-5887	519	21	m	m	PROPN
ejpam-5887	519	22	,	,	PUNCT
ejpam-5887	519	23	m	m	NOUN
ejpam-5887	519	24	)	)	PUNCT
ejpam-5887	519	25	)	)	PUNCT
ejpam-5887	520	1	=	=	PRON
ejpam-5887	520	2	{	{	PUNCT
ejpam-5887	520	3	vi	vi	PROPN
ejpam-5887	520	4	,	,	PUNCT
ejpam-5887	520	5	ui	ui	NOUN
ejpam-5887	520	6	,	,	PUNCT
ejpam-5887	520	7	vji	vji	VERB
ejpam-5887	520	8	,	,	PUNCT
ejpam-5887	520	9	u	u	NOUN
ejpam-5887	520	10	j	j	PROPN
ejpam-5887	521	1	i	i	PRON
ejpam-5887	521	2	:	:	PUNCT
ejpam-5887	521	3	1	1	NUM
ejpam-5887	521	4	≤	≤	NUM
ejpam-5887	521	5	i	i	PRON
ejpam-5887	521	6	≤	≤	PROPN
ejpam-5887	521	7	n	n	CCONJ
ejpam-5887	521	8	,	,	PUNCT
ejpam-5887	521	9	1	1	NUM
ejpam-5887	521	10	≤	≤	NUM
ejpam-5887	521	11	j	j	PROPN
ejpam-5887	521	12	≤	≤	PROPN
ejpam-5887	521	13	m	m	PROPN
ejpam-5887	521	14	}	}	PUNCT
ejpam-5887	521	15	and	and	CCONJ
ejpam-5887	521	16	e(p	e(p	PROPN
ejpam-5887	521	17	(	(	PUNCT
ejpam-5887	521	18	n.bv	n.bv	NOUN
ejpam-5887	521	19	m	m	PROPN
ejpam-5887	521	20	,	,	PUNCT
ejpam-5887	521	21	m	m	NOUN
ejpam-5887	521	22	)	)	PUNCT
ejpam-5887	521	23	)	)	PUNCT
ejpam-5887	522	1	=	=	PRON
ejpam-5887	522	2	{	{	PUNCT
ejpam-5887	522	3	vivi+1	vivi+1	PROPN
ejpam-5887	522	4	,	,	PUNCT
ejpam-5887	522	5	viui	viui	PROPN
ejpam-5887	522	6	,	,	PUNCT
ejpam-5887	522	7	viv	viv	PROPN
ejpam-5887	522	8	j	j	PROPN
ejpam-5887	522	9	i	i	PROPN
ejpam-5887	522	10	,	,	PUNCT
ejpam-5887	522	11	uiu	uiu	PROPN
ejpam-5887	522	12	j	j	PROPN
ejpam-5887	523	1	i	i	PROPN
ejpam-5887	523	2	,	,	PUNCT
ejpam-5887	523	3	vnun	vnun	PROPN
ejpam-5887	523	4	,	,	PUNCT
ejpam-5887	523	5	vnv	vnv	NOUN
ejpam-5887	523	6	j	j	PROPN
ejpam-5887	523	7	n	n	CCONJ
ejpam-5887	523	8	,	,	PUNCT
ejpam-5887	523	9	unu	unu	PROPN
ejpam-5887	523	10	j	j	PROPN
ejpam-5887	523	11	n	n	CCONJ
ejpam-5887	523	12	:	:	PUNCT
ejpam-5887	523	13	1	1	NUM
ejpam-5887	523	14	≤	≤	NUM
ejpam-5887	523	15	i	i	PRON
ejpam-5887	523	16	≤	≤	ADJ
ejpam-5887	523	17	n−	n−	PROPN
ejpam-5887	523	18	1	1	NUM
ejpam-5887	523	19	,	,	PUNCT
ejpam-5887	523	20	1	1	NUM
ejpam-5887	523	21	≤	≤	NUM
ejpam-5887	523	22	j	j	PROPN
ejpam-5887	523	23	≤	≤	PROPN
ejpam-5887	523	24	m	m	VERB
ejpam-5887	523	25	}	}	PUNCT
ejpam-5887	523	26	respectively	respectively	ADV
ejpam-5887	523	27	.	.	PUNCT
ejpam-5887	524	1	define	define	VERB
ejpam-5887	524	2	f	f	PROPN
ejpam-5887	524	3	:	:	PUNCT
ejpam-5887	524	4	e(p	e(p	PROPN
ejpam-5887	524	5	(	(	PUNCT
ejpam-5887	524	6	n.bv	n.bv	NOUN
ejpam-5887	524	7	m	m	PROPN
ejpam-5887	524	8	,	,	PUNCT
ejpam-5887	524	9	m	m	NOUN
ejpam-5887	524	10	)	)	PUNCT
ejpam-5887	524	11	)	)	PUNCT
ejpam-5887	525	1	→	→	PUNCT
ejpam-5887	525	2	{	{	PUNCT
ejpam-5887	525	3	0	0	NUM
ejpam-5887	525	4	,	,	PUNCT
ejpam-5887	525	5	1	1	NUM
ejpam-5887	525	6	,	,	PUNCT
ejpam-5887	525	7	2	2	NUM
ejpam-5887	525	8	,	,	PUNCT
ejpam-5887	525	9	...	...	PUNCT
ejpam-5887	525	10	,	,	PUNCT
ejpam-5887	525	11	k	k	PROPN
ejpam-5887	526	1	−	−	PROPN
ejpam-5887	526	2	1	1	NUM
ejpam-5887	526	3	}	}	PUNCT
ejpam-5887	526	4	for	for	ADP
ejpam-5887	526	5	n	n	PRON
ejpam-5887	526	6	≡	≡	PROPN
ejpam-5887	526	7	k	k	PROPN
ejpam-5887	527	1	−	−	PROPN
ejpam-5887	527	2	1	1	NUM
ejpam-5887	527	3	(	(	PUNCT
ejpam-5887	527	4	mod	mod	NOUN
ejpam-5887	527	5	k	k	PROPN
ejpam-5887	527	6	)	)	PUNCT
ejpam-5887	527	7	and	and	CCONJ
ejpam-5887	527	8	m	m	PROPN
ejpam-5887	527	9	≡	≡	PROPN
ejpam-5887	527	10	0	0	NUM
ejpam-5887	527	11	,	,	PUNCT
ejpam-5887	527	12	k	k	PROPN
ejpam-5887	527	13	−	−	PROPN
ejpam-5887	527	14	1	1	NUM
ejpam-5887	527	15	(	(	PUNCT
ejpam-5887	527	16	mod	mod	PROPN
ejpam-5887	527	17	k	k	PROPN
ejpam-5887	527	18	)	)	PUNCT
ejpam-5887	527	19	as	as	SCONJ
ejpam-5887	527	20	follows	follow	VERB
ejpam-5887	527	21	:	:	PUNCT
ejpam-5887	527	22	f(viv	f(viv	PROPN
ejpam-5887	528	1	j	j	PROPN
ejpam-5887	529	1	i	i	PROPN
ejpam-5887	529	2	)	)	PUNCT
ejpam-5887	529	3	;	;	PUNCT
ejpam-5887	529	4	1	1	NUM
ejpam-5887	529	5	≤	≤	NUM
ejpam-5887	530	1	i	i	PRON
ejpam-5887	530	2	≤	≤	ADJ
ejpam-5887	531	1	n−	n−	NOUN
ejpam-5887	531	2	k	k	PROPN
ejpam-5887	532	1	+	+	CCONJ
ejpam-5887	532	2	1	1	NUM
ejpam-5887	532	3	,	,	PUNCT
ejpam-5887	532	4	1	1	NUM
ejpam-5887	532	5	≤	≤	NUM
ejpam-5887	532	6	j	j	PROPN
ejpam-5887	532	7	≤	≤	NUM
ejpam-5887	532	8	m	m	VERB
ejpam-5887	532	9	as	as	ADP
ejpam-5887	532	10	in	in	ADP
ejpam-5887	532	11	case	case	NOUN
ejpam-5887	532	12	(	(	PUNCT
ejpam-5887	532	13	i	i	NOUN
ejpam-5887	532	14	)	)	PUNCT
ejpam-5887	532	15	of	of	ADP
ejpam-5887	532	16	theorem	theorem	NOUN
ejpam-5887	532	17	17	17	NUM
ejpam-5887	532	18	,	,	PUNCT
ejpam-5887	532	19	f(uiu	f(uiu	ADP
ejpam-5887	532	20	j	j	PROPN
ejpam-5887	532	21	i	i	PROPN
ejpam-5887	532	22	)	)	PUNCT
ejpam-5887	532	23	;	;	PUNCT
ejpam-5887	532	24	1	1	NUM
ejpam-5887	532	25	≤	≤	NUM
ejpam-5887	532	26	i	i	PRON
ejpam-5887	532	27	≤	≤	ADJ
ejpam-5887	533	1	n−	n−	NOUN
ejpam-5887	533	2	k	k	PROPN
ejpam-5887	534	1	+	+	CCONJ
ejpam-5887	534	2	1	1	NUM
ejpam-5887	534	3	,	,	PUNCT
ejpam-5887	534	4	1	1	NUM
ejpam-5887	534	5	≤	≤	NUM
ejpam-5887	534	6	j	j	PROPN
ejpam-5887	534	7	≤	≤	NUM
ejpam-5887	534	8	m	m	VERB
ejpam-5887	534	9	as	as	ADP
ejpam-5887	534	10	in	in	ADP
ejpam-5887	534	11	case	case	NOUN
ejpam-5887	534	12	(	(	PUNCT
ejpam-5887	534	13	i	i	NOUN
ejpam-5887	534	14	)	)	PUNCT
ejpam-5887	534	15	of	of	ADP
ejpam-5887	534	16	theorem	theorem	ADJ
ejpam-5887	534	17	17	17	NUM
ejpam-5887	534	18	,	,	PUNCT
ejpam-5887	534	19	f(vivi+1	f(vivi+1	NOUN
ejpam-5887	534	20	)	)	PUNCT
ejpam-5887	534	21	=	=	SYM
ejpam-5887	534	22	0	0	NUM
ejpam-5887	534	23	;	;	PUNCT
ejpam-5887	534	24	1	1	NUM
ejpam-5887	534	25	≤	≤	NUM
ejpam-5887	535	1	i	i	PRON
ejpam-5887	535	2	≤	≤	ADJ
ejpam-5887	535	3	n−	n−	PROPN
ejpam-5887	535	4	1	1	NUM
ejpam-5887	535	5	,	,	PUNCT
ejpam-5887	535	6	f(viui	f(viui	NOUN
ejpam-5887	535	7	)	)	PUNCT
ejpam-5887	535	8	=	=	SYM
ejpam-5887	535	9	0	0	NUM
ejpam-5887	535	10	;	;	PUNCT
ejpam-5887	535	11	1	1	NUM
ejpam-5887	535	12	≤	≤	NUM
ejpam-5887	535	13	i	i	PRON
ejpam-5887	535	14	≤	≤	PROPN
ejpam-5887	535	15	n	n	CCONJ
ejpam-5887	535	16	,	,	PUNCT
ejpam-5887	535	17	we	we	PRON
ejpam-5887	535	18	have	have	VERB
ejpam-5887	535	19	the	the	DET
ejpam-5887	535	20	following	follow	VERB
ejpam-5887	535	21	two	two	NUM
ejpam-5887	535	22	cases	case	NOUN
ejpam-5887	535	23	.	.	PUNCT
ejpam-5887	536	1	case	case	NOUN
ejpam-5887	536	2	(	(	PUNCT
ejpam-5887	536	3	i	i	NOUN
ejpam-5887	536	4	):	):	PUNCT
ejpam-5887	536	5	if	if	SCONJ
ejpam-5887	536	6	m	m	VERB
ejpam-5887	536	7	≡	≡	PROPN
ejpam-5887	536	8	0	0	PUNCT
ejpam-5887	537	1	(	(	PUNCT
ejpam-5887	537	2	mod	mod	PROPN
ejpam-5887	537	3	k	k	PROPN
ejpam-5887	537	4	)	)	PUNCT
ejpam-5887	538	1	,	,	PUNCT
ejpam-5887	538	2	then	then	ADV
ejpam-5887	538	3	f(viv	f(viv	PROPN
ejpam-5887	538	4	j	j	PROPN
ejpam-5887	538	5	i	i	NOUN
ejpam-5887	538	6	)	)	PUNCT
ejpam-5887	539	1	=	=	PUNCT
ejpam-5887	539	2	f(uiu	f(uiu	ADP
ejpam-5887	539	3	j	j	PROPN
ejpam-5887	539	4	i	i	NOUN
ejpam-5887	539	5	)	)	PUNCT
ejpam-5887	539	6	=	=	SYM
ejpam-5887	539	7	0	0	NUM
ejpam-5887	539	8	;	;	PUNCT
ejpam-5887	539	9	n−	n−	NOUN
ejpam-5887	539	10	k	k	NOUN
ejpam-5887	539	11	+	+	CCONJ
ejpam-5887	539	12	2	2	X
ejpam-5887	539	13	≤	≤	NUM
ejpam-5887	539	14	i	i	PRON
ejpam-5887	539	15	≤	≤	ADJ
ejpam-5887	539	16	n	n	CCONJ
ejpam-5887	539	17	,	,	PUNCT
ejpam-5887	539	18	1	1	NUM
ejpam-5887	539	19	≤	≤	NUM
ejpam-5887	539	20	j	j	PROPN
ejpam-5887	540	1	≤	≤	NUM
ejpam-5887	540	2	m	m	VERB
ejpam-5887	540	3	k	k	NOUN
ejpam-5887	540	4	−	−	PROPN
ejpam-5887	541	1	1	1	NUM
ejpam-5887	541	2	,	,	PUNCT
ejpam-5887	541	3	f(vn−k+1+iv	f(vn−k+1+iv	NOUN
ejpam-5887	541	4	m	m	NOUN
ejpam-5887	541	5	k	k	NOUN
ejpam-5887	541	6	n−k+1+i	n−k+1+i	NOUN
ejpam-5887	541	7	)	)	PUNCT
ejpam-5887	542	1	=	=	PUNCT
ejpam-5887	543	1	i	i	NOUN
ejpam-5887	543	2	;	;	PUNCT
ejpam-5887	543	3	1	1	NUM
ejpam-5887	543	4	≤	≤	NUM
ejpam-5887	543	5	i	i	X
ejpam-5887	543	6	≤	≤	NOUN
ejpam-5887	544	1	k	k	PRON
ejpam-5887	544	2	−	−	PROPN
ejpam-5887	544	3	2	2	NUM
ejpam-5887	544	4	,	,	PUNCT
ejpam-5887	544	5	f(vnv	f(vnv	NOUN
ejpam-5887	544	6	m	m	NOUN
ejpam-5887	544	7	k	k	NOUN
ejpam-5887	544	8	n	n	PROPN
ejpam-5887	544	9	)	)	PUNCT
ejpam-5887	544	10	=	=	PUNCT
ejpam-5887	545	1	f(unu	f(unu	ADJ
ejpam-5887	545	2	m	m	VERB
ejpam-5887	545	3	k	k	NOUN
ejpam-5887	545	4	n	n	PROPN
ejpam-5887	545	5	)	)	PUNCT
ejpam-5887	546	1	=	=	SYM
ejpam-5887	546	2	0	0	NUM
ejpam-5887	546	3	,	,	PUNCT
ejpam-5887	546	4	f(un−k+1+iu	f(un−k+1+iu	PROPN
ejpam-5887	546	5	m	m	PROPN
ejpam-5887	546	6	k	k	NOUN
ejpam-5887	546	7	n−k+1+i	n−k+1+i	PROPN
ejpam-5887	546	8	)	)	PUNCT
ejpam-5887	546	9	=	=	PUNCT
ejpam-5887	547	1	k	k	X
ejpam-5887	548	1	−	−	PROPN
ejpam-5887	549	1	i	i	PRON
ejpam-5887	549	2	;	;	PUNCT
ejpam-5887	549	3	1	1	NUM
ejpam-5887	549	4	≤	≤	NUM
ejpam-5887	549	5	i	i	X
ejpam-5887	549	6	≤	≤	NOUN
ejpam-5887	550	1	k	k	PRON
ejpam-5887	550	2	−	−	PROPN
ejpam-5887	550	3	2	2	NUM
ejpam-5887	550	4	,	,	PUNCT
ejpam-5887	550	5	f(viv	f(viv	PROPN
ejpam-5887	550	6	j	j	PROPN
ejpam-5887	551	1	i	i	NOUN
ejpam-5887	551	2	)	)	PUNCT
ejpam-5887	552	1	=	=	PUNCT
ejpam-5887	552	2	f(uiu	f(uiu	ADP
ejpam-5887	552	3	j	j	PROPN
ejpam-5887	552	4	i	i	NOUN
ejpam-5887	552	5	)	)	PUNCT
ejpam-5887	552	6	=	=	PUNCT
ejpam-5887	552	7			NOUN
ejpam-5887	552	8	1	1	NUM
ejpam-5887	552	9	;	;	PUNCT
ejpam-5887	552	10	m	m	VERB
ejpam-5887	552	11	k	k	NOUN
ejpam-5887	553	1	+	+	CCONJ
ejpam-5887	553	2	1	1	NUM
ejpam-5887	553	3	≤	≤	NUM
ejpam-5887	553	4	j	j	PROPN
ejpam-5887	553	5	≤	≤	ADV
ejpam-5887	553	6	2	2	NUM
ejpam-5887	553	7	m	m	NOUN
ejpam-5887	553	8	k	k	NOUN
ejpam-5887	553	9	2	2	NUM
ejpam-5887	553	10	;	;	PUNCT
ejpam-5887	553	11	2	2	NUM
ejpam-5887	553	12	m	m	NOUN
ejpam-5887	553	13	k	k	NOUN
ejpam-5887	553	14	+	+	CCONJ
ejpam-5887	553	15	1	1	NUM
ejpam-5887	553	16	≤	≤	NUM
ejpam-5887	553	17	j	j	PROPN
ejpam-5887	553	18	≤	≤	ADV
ejpam-5887	553	19	3	3	NUM
ejpam-5887	553	20	m	m	NOUN
ejpam-5887	553	21	k	k	NOUN
ejpam-5887	553	22	:	:	PUNCT
ejpam-5887	553	23	:	:	PUNCT
ejpam-5887	553	24	k	k	X
ejpam-5887	553	25	−	−	PROPN
ejpam-5887	553	26	1	1	NUM
ejpam-5887	553	27	;	;	PUNCT
ejpam-5887	553	28	(	(	PUNCT
ejpam-5887	553	29	k−1)m	k−1)m	PROPN
ejpam-5887	553	30	k	k	PROPN
ejpam-5887	553	31	+	+	CCONJ
ejpam-5887	553	32	1	1	NUM
ejpam-5887	553	33	≤	≤	NUM
ejpam-5887	553	34	j	j	PROPN
ejpam-5887	553	35	≤	≤	NUM
ejpam-5887	553	36	m	m	VERB
ejpam-5887	553	37	;	;	PUNCT
ejpam-5887	553	38	n−	n−	NOUN
ejpam-5887	553	39	k	k	NOUN
ejpam-5887	554	1	+	+	CCONJ
ejpam-5887	554	2	2	2	X
ejpam-5887	554	3	≤	≤	NUM
ejpam-5887	554	4	i	i	PRON
ejpam-5887	554	5	≤	≤	ADJ
ejpam-5887	554	6	n.	n.	NOUN
ejpam-5887	554	7	from	from	ADP
ejpam-5887	554	8	this	this	DET
ejpam-5887	554	9	labeling	labeling	NOUN
ejpam-5887	554	10	we	we	PRON
ejpam-5887	554	11	have	have	VERB
ejpam-5887	554	12	,	,	PUNCT
ejpam-5887	554	13	ef	ef	PROPN
ejpam-5887	554	14	(	(	PUNCT
ejpam-5887	554	15	i	i	NOUN
ejpam-5887	554	16	)	)	PUNCT
ejpam-5887	554	17	=	=	PRON
ejpam-5887	554	18	{	{	PUNCT
ejpam-5887	554	19	2⌊nk	2⌊nk	NUM
ejpam-5887	554	20	⌋+	⌋+	PUNCT
ejpam-5887	555	1	2mn	2mn	PROPN
ejpam-5887	555	2	k	k	X
ejpam-5887	556	1	+	+	CCONJ
ejpam-5887	556	2	1	1	NUM
ejpam-5887	556	3	;	;	PUNCT
ejpam-5887	556	4	i	i	PROPN
ejpam-5887	556	5	=	=	NOUN
ejpam-5887	556	6	0	0	NUM
ejpam-5887	556	7	,	,	PUNCT
ejpam-5887	556	8	1	1	NUM
ejpam-5887	556	9	,	,	PUNCT
ejpam-5887	556	10	k	k	PROPN
ejpam-5887	557	1	−	−	PROPN
ejpam-5887	557	2	1	1	NUM
ejpam-5887	557	3	2⌊nk	2⌊nk	NUM
ejpam-5887	557	4	⌋+	⌋+	PUNCT
ejpam-5887	558	1	2mn	2mn	PROPN
ejpam-5887	558	2	k	k	X
ejpam-5887	559	1	+	+	CCONJ
ejpam-5887	559	2	2	2	NUM
ejpam-5887	559	3	;	;	PUNCT
ejpam-5887	559	4	2	2	NUM
ejpam-5887	559	5	≤	≤	NUM
ejpam-5887	559	6	i	i	X
ejpam-5887	559	7	≤	≤	PUNCT
ejpam-5887	560	1	k	k	PRON
ejpam-5887	560	2	−	−	PROPN
ejpam-5887	560	3	2	2	NUM
ejpam-5887	560	4	,	,	PUNCT
ejpam-5887	560	5	n.	n.	NOUN
ejpam-5887	560	6	m.	m.	NOUN
ejpam-5887	560	7	noureldeen	noureldeen	NOUN
ejpam-5887	560	8	et	et	PROPN
ejpam-5887	560	9	al	al	PROPN
ejpam-5887	560	10	.	.	PUNCT
ejpam-5887	560	11	/	/	SYM
ejpam-5887	560	12	eur	eur	PROPN
ejpam-5887	560	13	.	.	PUNCT
ejpam-5887	561	1	j.	j.	PROPN
ejpam-5887	561	2	pure	pure	PROPN
ejpam-5887	561	3	appl	appl	PROPN
ejpam-5887	561	4	.	.	PROPN
ejpam-5887	561	5	math	math	PROPN
ejpam-5887	561	6	,	,	PUNCT
ejpam-5887	561	7	18	18	NUM
ejpam-5887	561	8	(	(	PUNCT
ejpam-5887	561	9	2	2	NUM
ejpam-5887	561	10	)	)	PUNCT
ejpam-5887	561	11	(	(	PUNCT
ejpam-5887	561	12	2025	2025	NUM
ejpam-5887	561	13	)	)	PUNCT
ejpam-5887	561	14	,	,	PUNCT
ejpam-5887	561	15	5887	5887	NUM
ejpam-5887	561	16	16	16	NUM
ejpam-5887	561	17	of	of	ADP
ejpam-5887	561	18	21	21	NUM
ejpam-5887	561	19	vf∗(i	vf∗(i	ADJ
ejpam-5887	561	20	)	)	PUNCT
ejpam-5887	561	21	=	=	PUNCT
ejpam-5887	561	22	{	{	PUNCT
ejpam-5887	561	23	2⌊nk	2⌊nk	NUM
ejpam-5887	561	24	⌋+	⌋+	PUNCT
ejpam-5887	562	1	2mn	2mn	PROPN
ejpam-5887	562	2	k	k	X
ejpam-5887	563	1	+	+	CCONJ
ejpam-5887	563	2	1	1	NUM
ejpam-5887	563	3	;	;	PUNCT
ejpam-5887	563	4	i	i	PRON
ejpam-5887	563	5	=	=	NOUN
ejpam-5887	563	6	1	1	NUM
ejpam-5887	563	7	,	,	PUNCT
ejpam-5887	563	8	k	k	PROPN
ejpam-5887	564	1	−	−	PROPN
ejpam-5887	564	2	1	1	NUM
ejpam-5887	564	3	2⌊nk	2⌊nk	NUM
ejpam-5887	564	4	⌋+	⌋+	PUNCT
ejpam-5887	565	1	2mn	2mn	PROPN
ejpam-5887	565	2	k	k	X
ejpam-5887	566	1	+	+	CCONJ
ejpam-5887	566	2	2	2	NUM
ejpam-5887	566	3	;	;	PUNCT
ejpam-5887	566	4	i	i	PROPN
ejpam-5887	566	5	=	=	NOUN
ejpam-5887	566	6	0	0	NUM
ejpam-5887	566	7	;	;	PUNCT
ejpam-5887	566	8	2	2	NUM
ejpam-5887	566	9	≤	≤	NUM
ejpam-5887	566	10	i	i	X
ejpam-5887	566	11	≤	≤	PUNCT
ejpam-5887	567	1	k	k	PRON
ejpam-5887	567	2	−	−	NOUN
ejpam-5887	567	3	2	2	X
ejpam-5887	567	4	.	.	PUNCT
ejpam-5887	567	5	case	case	NOUN
ejpam-5887	567	6	(	(	PUNCT
ejpam-5887	567	7	ii	ii	NOUN
ejpam-5887	567	8	):	):	PUNCT
ejpam-5887	567	9	if	if	SCONJ
ejpam-5887	567	10	m	m	VERB
ejpam-5887	568	1	≡	≡	PROPN
ejpam-5887	568	2	k	k	PROPN
ejpam-5887	569	1	−	−	PROPN
ejpam-5887	569	2	1	1	NUM
ejpam-5887	569	3	(	(	PUNCT
ejpam-5887	569	4	mod	mod	PROPN
ejpam-5887	569	5	k	k	PROPN
ejpam-5887	569	6	)	)	PUNCT
ejpam-5887	569	7	,	,	PUNCT
ejpam-5887	569	8	then	then	ADV
ejpam-5887	569	9	f(viv	f(viv	PROPN
ejpam-5887	569	10	j	j	PROPN
ejpam-5887	569	11	i	i	PROPN
ejpam-5887	569	12	)	)	PUNCT
ejpam-5887	569	13	;	;	PUNCT
ejpam-5887	569	14	n−	n−	NOUN
ejpam-5887	569	15	k	k	NOUN
ejpam-5887	570	1	+	+	CCONJ
ejpam-5887	570	2	2	2	X
ejpam-5887	570	3	≤	≤	NUM
ejpam-5887	570	4	i	i	PRON
ejpam-5887	570	5	≤	≤	ADJ
ejpam-5887	570	6	n	n	CCONJ
ejpam-5887	570	7	,	,	PUNCT
ejpam-5887	570	8	1	1	NUM
ejpam-5887	570	9	≤	≤	NUM
ejpam-5887	570	10	j	j	PROPN
ejpam-5887	570	11	≤	≤	PROPN
ejpam-5887	570	12	m−	m−	PROPN
ejpam-5887	570	13	k	k	PROPN
ejpam-5887	571	1	+	+	CCONJ
ejpam-5887	571	2	1	1	NUM
ejpam-5887	571	3	as	as	ADP
ejpam-5887	571	4	in	in	ADP
ejpam-5887	571	5	case	case	NOUN
ejpam-5887	571	6	(	(	PUNCT
ejpam-5887	571	7	i	i	NOUN
ejpam-5887	571	8	)	)	PUNCT
ejpam-5887	571	9	,	,	PUNCT
ejpam-5887	571	10	f(uiu	f(uiu	ADP
ejpam-5887	571	11	j	j	PROPN
ejpam-5887	571	12	i	i	PROPN
ejpam-5887	571	13	)	)	PUNCT
ejpam-5887	571	14	;	;	PUNCT
ejpam-5887	571	15	n−	n−	NOUN
ejpam-5887	571	16	k	k	NOUN
ejpam-5887	571	17	+	+	CCONJ
ejpam-5887	571	18	2	2	X
ejpam-5887	571	19	≤	≤	NUM
ejpam-5887	571	20	i	i	PRON
ejpam-5887	571	21	≤	≤	ADJ
ejpam-5887	571	22	n	n	CCONJ
ejpam-5887	571	23	,	,	PUNCT
ejpam-5887	571	24	1	1	NUM
ejpam-5887	571	25	≤	≤	NUM
ejpam-5887	571	26	j	j	PROPN
ejpam-5887	571	27	≤	≤	PROPN
ejpam-5887	572	1	m−	m−	PROPN
ejpam-5887	572	2	k	k	PROPN
ejpam-5887	573	1	+	+	CCONJ
ejpam-5887	573	2	1	1	NUM
ejpam-5887	573	3	as	as	ADP
ejpam-5887	573	4	in	in	ADP
ejpam-5887	573	5	case	case	NOUN
ejpam-5887	573	6	(	(	PUNCT
ejpam-5887	573	7	i	i	NOUN
ejpam-5887	573	8	)	)	PUNCT
ejpam-5887	573	9	,	,	PUNCT
ejpam-5887	573	10	f(vivi+1	f(vivi+1	NOUN
ejpam-5887	573	11	)	)	PUNCT
ejpam-5887	573	12	=	=	SYM
ejpam-5887	573	13	0	0	NUM
ejpam-5887	573	14	;	;	PUNCT
ejpam-5887	573	15	1	1	NUM
ejpam-5887	573	16	≤	≤	NUM
ejpam-5887	574	1	i	i	PRON
ejpam-5887	574	2	≤	≤	ADJ
ejpam-5887	574	3	n−	n−	PROPN
ejpam-5887	574	4	1	1	NUM
ejpam-5887	574	5	,	,	PUNCT
ejpam-5887	574	6	f(viui	f(viui	NOUN
ejpam-5887	574	7	)	)	PUNCT
ejpam-5887	574	8	=	=	SYM
ejpam-5887	574	9	0	0	NUM
ejpam-5887	574	10	;	;	PUNCT
ejpam-5887	574	11	1	1	NUM
ejpam-5887	574	12	≤	≤	NUM
ejpam-5887	574	13	i	i	PRON
ejpam-5887	574	14	≤	≤	PROPN
ejpam-5887	574	15	n	n	CCONJ
ejpam-5887	574	16	,	,	PUNCT
ejpam-5887	574	17	f(viv	f(viv	PROPN
ejpam-5887	574	18	j	j	PROPN
ejpam-5887	575	1	i	i	NOUN
ejpam-5887	575	2	)	)	PUNCT
ejpam-5887	576	1	=	=	SYM
ejpam-5887	576	2	0	0	NUM
ejpam-5887	576	3	;	;	PUNCT
ejpam-5887	576	4	n−	n−	NOUN
ejpam-5887	576	5	k	k	NOUN
ejpam-5887	577	1	+	+	CCONJ
ejpam-5887	577	2	2	2	X
ejpam-5887	577	3	≤	≤	NUM
ejpam-5887	577	4	i	i	PRON
ejpam-5887	577	5	≤	≤	ADJ
ejpam-5887	577	6	n	n	X
ejpam-5887	577	7	,	,	PUNCT
ejpam-5887	577	8	m−	m−	PROPN
ejpam-5887	577	9	k	k	PROPN
ejpam-5887	578	1	+	+	CCONJ
ejpam-5887	578	2	2	2	NUM
ejpam-5887	578	3	≤	≤	NUM
ejpam-5887	578	4	j	j	PROPN
ejpam-5887	578	5	≤	≤	PROPN
ejpam-5887	579	1	m−	m−	PROPN
ejpam-5887	579	2	k	k	PROPN
ejpam-5887	580	1	+	+	CCONJ
ejpam-5887	580	2	⌊mk	⌊mk	X
ejpam-5887	580	3	⌋+1	⌋+1	PROPN
ejpam-5887	580	4	,	,	PUNCT
ejpam-5887	580	5	f(vn−k+1+iv	f(vn−k+1+iv	NOUN
ejpam-5887	580	6	j	j	PROPN
ejpam-5887	580	7	n−k+1+i	n−k+1+i	NOUN
ejpam-5887	580	8	)	)	PUNCT
ejpam-5887	580	9	=	=	PUNCT
ejpam-5887	580	10	0	0	NUM
ejpam-5887	580	11	;	;	PUNCT
ejpam-5887	581	1	2	2	NUM
ejpam-5887	581	2	≤	≤	NUM
ejpam-5887	581	3	i	i	X
ejpam-5887	581	4	≤	≤	PUNCT
ejpam-5887	582	1	k	k	PRON
ejpam-5887	583	1	−	−	PROPN
ejpam-5887	583	2	2	2	NUM
ejpam-5887	583	3	,	,	PUNCT
ejpam-5887	583	4	m−	m−	PROPN
ejpam-5887	583	5	k	k	PROPN
ejpam-5887	584	1	+	+	CCONJ
ejpam-5887	584	2	⌊mk	⌊mk	X
ejpam-5887	584	3	⌋+	⌋+	X
ejpam-5887	584	4	2	2	NUM
ejpam-5887	584	5	≤	≤	NUM
ejpam-5887	584	6	j	j	PROPN
ejpam-5887	584	7	≤	≤	PROPN
ejpam-5887	585	1	m−	m−	PROPN
ejpam-5887	585	2	k	k	PROPN
ejpam-5887	586	1	+	+	PROPN
ejpam-5887	586	2	2⌊mk	2⌊mk	NUM
ejpam-5887	586	3	⌋+	⌋+	NUM
ejpam-5887	586	4	1	1	NUM
ejpam-5887	586	5	,	,	PUNCT
ejpam-5887	586	6	f(vn−k+1+iv	f(vn−k+1+iv	NOUN
ejpam-5887	586	7	j	j	PROPN
ejpam-5887	586	8	n−k+1+i	n−k+1+i	NOUN
ejpam-5887	586	9	)	)	PUNCT
ejpam-5887	587	1	=	=	PUNCT
ejpam-5887	588	1	i	i	NOUN
ejpam-5887	588	2	;	;	PUNCT
ejpam-5887	588	3	1	1	NUM
ejpam-5887	588	4	≤	≤	NUM
ejpam-5887	588	5	i	i	X
ejpam-5887	588	6	≤	≤	PUNCT
ejpam-5887	589	1	k	k	PRON
ejpam-5887	590	1	−	−	PROPN
ejpam-5887	590	2	1	1	NUM
ejpam-5887	590	3	,	,	PUNCT
ejpam-5887	590	4	m−	m−	PROPN
ejpam-5887	590	5	k	k	PROPN
ejpam-5887	591	1	+	+	PROPN
ejpam-5887	591	2	2⌊mk	2⌊mk	NUM
ejpam-5887	591	3	⌋+	⌋+	NUM
ejpam-5887	591	4	2	2	NUM
ejpam-5887	591	5	≤	≤	NUM
ejpam-5887	591	6	j	j	PROPN
ejpam-5887	591	7	≤	≤	NUM
ejpam-5887	591	8	m	m	PROPN
ejpam-5887	591	9	,	,	PUNCT
ejpam-5887	591	10	f(vn−k+1+iv	f(vn−k+1+iv	NOUN
ejpam-5887	591	11	j	j	PROPN
ejpam-5887	591	12	n−k+1+i	n−k+1+i	NOUN
ejpam-5887	591	13	)	)	PUNCT
ejpam-5887	592	1	=	=	PUNCT
ejpam-5887	593	1	i	i	INTJ
ejpam-5887	593	2	;	;	PUNCT
ejpam-5887	593	3	i	i	NOUN
ejpam-5887	593	4	=	=	NOUN
ejpam-5887	593	5	1	1	NUM
ejpam-5887	593	6	,	,	PUNCT
ejpam-5887	593	7	k	k	PROPN
ejpam-5887	594	1	−	−	PROPN
ejpam-5887	594	2	1	1	NUM
ejpam-5887	594	3	,	,	PUNCT
ejpam-5887	594	4	m−	m−	PROPN
ejpam-5887	594	5	k	k	PROPN
ejpam-5887	595	1	+	+	CCONJ
ejpam-5887	595	2	2	2	NUM
ejpam-5887	595	3	≤	≤	NUM
ejpam-5887	595	4	j	j	PROPN
ejpam-5887	595	5	≤	≤	PROPN
ejpam-5887	596	1	m−	m−	PROPN
ejpam-5887	596	2	k	k	PROPN
ejpam-5887	597	1	+	+	CCONJ
ejpam-5887	597	2	⌊mk	⌊mk	X
ejpam-5887	597	3	⌋+	⌋+	X
ejpam-5887	597	4	1	1	NUM
ejpam-5887	597	5	,	,	PUNCT
ejpam-5887	597	6	f(un−k+1+iu	f(un−k+1+iu	PROPN
ejpam-5887	597	7	j	j	PROPN
ejpam-5887	597	8	n−k+1+i	n−k+1+i	PROPN
ejpam-5887	597	9	)	)	PUNCT
ejpam-5887	598	1	=	=	PUNCT
ejpam-5887	599	1	i	i	NOUN
ejpam-5887	599	2	;	;	PUNCT
ejpam-5887	599	3	1	1	NUM
ejpam-5887	599	4	≤	≤	X
ejpam-5887	599	5	i	i	PRON
ejpam-5887	599	6	,	,	PUNCT
ejpam-5887	599	7	j	j	PROPN
ejpam-5887	599	8	≤	≤	PROPN
ejpam-5887	600	1	k	k	INTJ
ejpam-5887	601	1	−	−	PROPN
ejpam-5887	601	2	1	1	NUM
ejpam-5887	601	3	.	.	PUNCT
ejpam-5887	601	4	from	from	ADP
ejpam-5887	601	5	this	this	DET
ejpam-5887	601	6	labeling	labeling	NOUN
ejpam-5887	601	7	we	we	PRON
ejpam-5887	601	8	have	have	VERB
ejpam-5887	601	9	,	,	PUNCT
ejpam-5887	601	10	ef	ef	PROPN
ejpam-5887	601	11	(	(	PUNCT
ejpam-5887	601	12	i	i	NOUN
ejpam-5887	601	13	)	)	PUNCT
ejpam-5887	601	14	=	=	VERB
ejpam-5887	601	15	{	{	PUNCT
ejpam-5887	601	16	2n⌊mk	2n⌊mk	NUM
ejpam-5887	601	17	⌋+	⌋+	X
ejpam-5887	602	1	2n−	2n−	NUM
ejpam-5887	602	2	1	1	NUM
ejpam-5887	602	3	;	;	PUNCT
ejpam-5887	602	4	i	i	PRON
ejpam-5887	602	5	=	=	NOUN
ejpam-5887	602	6	0	0	NUM
ejpam-5887	602	7	2n⌊mk	2n⌊mk	NUM
ejpam-5887	602	8	⌋+	⌋+	NUM
ejpam-5887	602	9	2n	2n	NUM
ejpam-5887	602	10	;	;	PUNCT
ejpam-5887	602	11	1	1	NUM
ejpam-5887	602	12	≤	≤	NUM
ejpam-5887	602	13	i	i	X
ejpam-5887	602	14	≤	≤	NOUN
ejpam-5887	603	1	k	k	PRON
ejpam-5887	603	2	−	−	PROPN
ejpam-5887	603	3	1	1	NUM
ejpam-5887	603	4	,	,	PUNCT
ejpam-5887	603	5	vf∗(i	vf∗(i	ADJ
ejpam-5887	603	6	)	)	PUNCT
ejpam-5887	603	7	=	=	SYM
ejpam-5887	604	1	2n⌊mk	2n⌊mk	NUM
ejpam-5887	604	2	⌋+	⌋+	NUM
ejpam-5887	604	3	2n	2n	NUM
ejpam-5887	604	4	;	;	PUNCT
ejpam-5887	604	5	0	0	NUM
ejpam-5887	604	6	≤	≤	NUM
ejpam-5887	605	1	i	i	PRON
ejpam-5887	605	2	≤	≤	NOUN
ejpam-5887	606	1	k	k	PRON
ejpam-5887	607	1	−	−	NOUN
ejpam-5887	607	2	1	1	X
ejpam-5887	607	3	.	.	PUNCT
ejpam-5887	608	1	clearly	clearly	ADV
ejpam-5887	608	2	,	,	PUNCT
ejpam-5887	608	3	|ef	|ef	PROPN
ejpam-5887	608	4	(	(	PUNCT
ejpam-5887	608	5	i	i	NOUN
ejpam-5887	608	6	)	)	PUNCT
ejpam-5887	608	7	−	−	PROPN
ejpam-5887	608	8	ef	ef	PROPN
ejpam-5887	608	9	(	(	PUNCT
ejpam-5887	608	10	j)|	j)|	PROPN
ejpam-5887	608	11	≤	≤	NUM
ejpam-5887	608	12	1	1	NUM
ejpam-5887	608	13	and	and	CCONJ
ejpam-5887	608	14	|vf∗(i	|vf∗(i	NUM
ejpam-5887	608	15	)	)	PUNCT
ejpam-5887	608	16	−	−	NOUN
ejpam-5887	608	17	vf∗(j)|	vf∗(j)|	VERB
ejpam-5887	608	18	≤	≤	NOUN
ejpam-5887	608	19	1	1	NUM
ejpam-5887	608	20	for	for	ADP
ejpam-5887	608	21	i	i	PRON
ejpam-5887	608	22	,	,	PUNCT
ejpam-5887	608	23	j	j	PROPN
ejpam-5887	608	24	∈	∈	PROPN
ejpam-5887	608	25	{	{	PUNCT
ejpam-5887	608	26	0	0	NUM
ejpam-5887	608	27	,	,	PUNCT
ejpam-5887	608	28	1	1	NUM
ejpam-5887	608	29	,	,	PUNCT
ejpam-5887	608	30	...	...	PUNCT
ejpam-5887	608	31	,	,	PUNCT
ejpam-5887	608	32	k	k	PROPN
ejpam-5887	608	33	−	−	PROPN
ejpam-5887	608	34	1	1	NUM
ejpam-5887	608	35	}	}	PUNCT
ejpam-5887	608	36	.	.	PUNCT
ejpam-5887	609	1	hence	hence	ADV
ejpam-5887	609	2	,	,	PUNCT
ejpam-5887	609	3	p	p	X
ejpam-5887	609	4	(	(	PUNCT
ejpam-5887	609	5	n.bv	n.bv	NOUN
ejpam-5887	609	6	m	m	PROPN
ejpam-5887	609	7	,	,	PUNCT
ejpam-5887	609	8	m	m	VERB
ejpam-5887	609	9	)	)	PUNCT
ejpam-5887	609	10	is	be	AUX
ejpam-5887	609	11	an	an	DET
ejpam-5887	609	12	edge	edge	NOUN
ejpam-5887	609	13	k	k	NOUN
ejpam-5887	609	14	-	-	PUNCT
ejpam-5887	609	15	product	product	NOUN
ejpam-5887	609	16	cordial	cordial	ADJ
ejpam-5887	609	17	graph	graph	NOUN
ejpam-5887	609	18	if	if	SCONJ
ejpam-5887	609	19	n	n	PRON
ejpam-5887	609	20	≡	≡	PROPN
ejpam-5887	609	21	k	k	PROPN
ejpam-5887	610	1	−	−	PROPN
ejpam-5887	610	2	1	1	NUM
ejpam-5887	610	3	(	(	PUNCT
ejpam-5887	610	4	mod	mod	NOUN
ejpam-5887	610	5	k	k	PROPN
ejpam-5887	610	6	)	)	PUNCT
ejpam-5887	610	7	and	and	CCONJ
ejpam-5887	610	8	m	m	PROPN
ejpam-5887	610	9	≡	≡	PROPN
ejpam-5887	610	10	0	0	NUM
ejpam-5887	610	11	,	,	PUNCT
ejpam-5887	610	12	k	k	PROPN
ejpam-5887	610	13	−	−	PROPN
ejpam-5887	610	14	1	1	NUM
ejpam-5887	610	15	(	(	PUNCT
ejpam-5887	610	16	mod	mod	PROPN
ejpam-5887	610	17	k	k	PROPN
ejpam-5887	610	18	)	)	PUNCT
ejpam-5887	610	19	.	.	PUNCT
ejpam-5887	611	1	example	example	NOUN
ejpam-5887	612	1	6	6	NUM
ejpam-5887	612	2	.	.	PUNCT
ejpam-5887	613	1	an	an	DET
ejpam-5887	613	2	edge	edge	NOUN
ejpam-5887	613	3	3	3	NUM
ejpam-5887	613	4	-	-	PUNCT
ejpam-5887	613	5	product	product	NOUN
ejpam-5887	613	6	cordial	cordial	ADJ
ejpam-5887	613	7	labeling	labeling	NOUN
ejpam-5887	613	8	of	of	ADP
ejpam-5887	613	9	p	p	PROPN
ejpam-5887	613	10	(	(	PUNCT
ejpam-5887	613	11	5.bv	5.bv	NUM
ejpam-5887	613	12	3,3	3,3	NUM
ejpam-5887	613	13	)	)	PUNCT
ejpam-5887	613	14	is	be	AUX
ejpam-5887	613	15	given	give	VERB
ejpam-5887	613	16	in	in	ADP
ejpam-5887	613	17	figure	figure	NOUN
ejpam-5887	613	18	6	6	NUM
ejpam-5887	613	19	.	.	NOUN
ejpam-5887	613	20	0	0	NUM
ejpam-5887	613	21	0	0	NUM
ejpam-5887	613	22	0	0	NUM
ejpam-5887	613	23	00	00	NUM
ejpam-5887	613	24	0	0	NUM
ejpam-5887	613	25	0	0	NUM
ejpam-5887	613	26	0	0	NUM
ejpam-5887	613	27	0	0	NUM
ejpam-5887	613	28	0	0	NUM
ejpam-5887	613	29	0	0	NUM
ejpam-5887	613	30	0	0	NUM
ejpam-5887	613	31	0	0	NUM
ejpam-5887	613	32	0	0	NUM
ejpam-5887	613	33	0	0	NUM
ejpam-5887	613	34	0	0	NUM
ejpam-5887	613	35	0	0	NUM
ejpam-5887	613	36	0	0	NUM
ejpam-5887	613	37	0	0	NUM
ejpam-5887	613	38	0	0	NUM
ejpam-5887	613	39	0	0	NUM
ejpam-5887	613	40	0	0	NUM
ejpam-5887	613	41	0	0	NUM
ejpam-5887	613	42	0	0	NUM
ejpam-5887	613	43	0	0	NUM
ejpam-5887	613	44	0	0	NUM
ejpam-5887	613	45	0	0	NUM
ejpam-5887	613	46	1	1	NUM
ejpam-5887	613	47	1	1	NUM
ejpam-5887	613	48	1	1	NUM
ejpam-5887	613	49	1	1	NUM
ejpam-5887	613	50	1	1	NUM
ejpam-5887	613	51	1	1	NUM
ejpam-5887	613	52	1	1	NUM
ejpam-5887	613	53	1	1	NUM
ejpam-5887	613	54	1	1	NUM
ejpam-5887	613	55	1	1	NUM
ejpam-5887	613	56	1	1	NUM
ejpam-5887	613	57	1	1	NUM
ejpam-5887	613	58	1	1	NUM
ejpam-5887	613	59	1	1	NUM
ejpam-5887	613	60	1	1	NUM
ejpam-5887	613	61	1	1	NUM
ejpam-5887	613	62	1	1	NUM
ejpam-5887	613	63	1	1	NUM
ejpam-5887	613	64	1	1	NUM
ejpam-5887	613	65	1	1	NUM
ejpam-5887	613	66	1	1	NUM
ejpam-5887	613	67	1	1	NUM
ejpam-5887	613	68	1	1	NUM
ejpam-5887	613	69	1	1	NUM
ejpam-5887	613	70	1	1	NUM
ejpam-5887	613	71	1	1	NUM
ejpam-5887	613	72	2	2	NUM
ejpam-5887	613	73	2	2	NUM
ejpam-5887	613	74	2	2	NUM
ejpam-5887	613	75	2	2	NUM
ejpam-5887	613	76	2	2	NUM
ejpam-5887	613	77	2	2	NUM
ejpam-5887	613	78	2	2	NUM
ejpam-5887	613	79	2	2	NUM
ejpam-5887	613	80	2	2	NUM
ejpam-5887	613	81	2	2	NUM
ejpam-5887	613	82	2	2	NUM
ejpam-5887	613	83	2	2	NUM
ejpam-5887	613	84	2	2	NUM
ejpam-5887	613	85	2	2	NUM
ejpam-5887	613	86	2	2	NUM
ejpam-5887	613	87	2	2	NUM
ejpam-5887	613	88	2	2	NUM
ejpam-5887	613	89	2	2	NUM
ejpam-5887	613	90	2	2	NUM
ejpam-5887	613	91	2	2	NUM
ejpam-5887	613	92	2	2	NUM
ejpam-5887	613	93	2	2	NUM
ejpam-5887	613	94	2	2	NUM
ejpam-5887	613	95	2	2	NUM
ejpam-5887	613	96	2	2	NUM
ejpam-5887	613	97	2	2	NUM
ejpam-5887	613	98	figure	figure	NOUN
ejpam-5887	613	99	6	6	NUM
ejpam-5887	613	100	:	:	PUNCT
ejpam-5887	613	101	edge	edge	VERB
ejpam-5887	613	102	3	3	NUM
ejpam-5887	613	103	-	-	PUNCT
ejpam-5887	613	104	product	product	NOUN
ejpam-5887	613	105	cordial	cordial	ADJ
ejpam-5887	613	106	labeling	labeling	NOUN
ejpam-5887	613	107	of	of	ADP
ejpam-5887	613	108	p	p	PROPN
ejpam-5887	613	109	(	(	PUNCT
ejpam-5887	613	110	5.bv	5.bv	NUM
ejpam-5887	613	111	3,3	3,3	NUM
ejpam-5887	613	112	)	)	PUNCT
ejpam-5887	613	113	4.3	4.3	NUM
ejpam-5887	613	114	.	.	PUNCT
ejpam-5887	613	115	path	path	PROPN
ejpam-5887	613	116	union	union	PROPN
ejpam-5887	613	117	of	of	ADP
ejpam-5887	613	118	cycle	cycle	NOUN
ejpam-5887	613	119	in	in	ADP
ejpam-5887	613	120	this	this	DET
ejpam-5887	613	121	subsection	subsection	NOUN
ejpam-5887	613	122	,	,	PUNCT
ejpam-5887	613	123	we	we	PRON
ejpam-5887	613	124	establish	establish	VERB
ejpam-5887	613	125	the	the	DET
ejpam-5887	613	126	necessary	necessary	ADJ
ejpam-5887	613	127	conditions	condition	NOUN
ejpam-5887	613	128	for	for	ADP
ejpam-5887	613	129	the	the	DET
ejpam-5887	613	130	path	path	NOUN
ejpam-5887	613	131	union	union	NOUN
ejpam-5887	613	132	of	of	ADP
ejpam-5887	613	133	a	a	DET
ejpam-5887	613	134	cycle	cycle	NOUN
ejpam-5887	613	135	graph	graph	NOUN
ejpam-5887	613	136	p	p	X
ejpam-5887	613	137	(	(	PUNCT
ejpam-5887	613	138	n.cv	n.cv	INTJ
ejpam-5887	613	139	m	m	PROPN
ejpam-5887	613	140	)	)	PUNCT
ejpam-5887	613	141	to	to	PART
ejpam-5887	613	142	admit	admit	VERB
ejpam-5887	613	143	an	an	DET
ejpam-5887	613	144	edge	edge	NOUN
ejpam-5887	613	145	k	k	NOUN
ejpam-5887	613	146	-	-	PUNCT
ejpam-5887	613	147	product	product	NOUN
ejpam-5887	613	148	cordial	cordial	ADJ
ejpam-5887	613	149	labeling	labeling	NOUN
ejpam-5887	613	150	.	.	PUNCT
ejpam-5887	614	1	also	also	ADV
ejpam-5887	614	2	,	,	PUNCT
ejpam-5887	614	3	we	we	PRON
ejpam-5887	614	4	investigate	investigate	VERB
ejpam-5887	614	5	the	the	DET
ejpam-5887	614	6	edge	edge	NOUN
ejpam-5887	614	7	3	3	NUM
ejpam-5887	614	8	-	-	PUNCT
ejpam-5887	614	9	product	product	NOUN
ejpam-5887	614	10	and	and	CCONJ
ejpam-5887	614	11	4	4	NUM
ejpam-5887	614	12	-	-	PUNCT
ejpam-5887	614	13	product	product	NOUN
ejpam-5887	614	14	cordial	cordial	ADJ
ejpam-5887	614	15	behavior	behavior	NOUN
ejpam-5887	614	16	of	of	ADP
ejpam-5887	614	17	p	p	PROPN
ejpam-5887	614	18	(	(	PUNCT
ejpam-5887	614	19	n.cv	n.cv	INTJ
ejpam-5887	614	20	m	m	PROPN
ejpam-5887	614	21	)	)	PUNCT
ejpam-5887	614	22	.	.	PUNCT
ejpam-5887	615	1	let	let	VERB
ejpam-5887	615	2	v	v	PART
ejpam-5887	615	3	be	be	AUX
ejpam-5887	615	4	a	a	DET
ejpam-5887	615	5	vertex	vertex	NOUN
ejpam-5887	615	6	of	of	ADP
ejpam-5887	615	7	a	a	DET
ejpam-5887	615	8	cycle	cycle	NOUN
ejpam-5887	615	9	cm	cm	NOUN
ejpam-5887	615	10	;	;	PUNCT
ejpam-5887	615	11	m	m	VERB
ejpam-5887	615	12	≥	≥	NOUN
ejpam-5887	615	13	3	3	NUM
ejpam-5887	615	14	.	.	PUNCT
ejpam-5887	616	1	according	accord	VERB
ejpam-5887	616	2	to	to	ADP
ejpam-5887	616	3	the	the	DET
ejpam-5887	616	4	symmetry	symmetry	NOUN
ejpam-5887	616	5	,	,	PUNCT
ejpam-5887	616	6	all	all	DET
ejpam-5887	616	7	p	p	X
ejpam-5887	616	8	(	(	PUNCT
ejpam-5887	616	9	n.cv	n.cv	PROPN
ejpam-5887	616	10	m	m	PROPN
ejpam-5887	616	11	)	)	PUNCT
ejpam-5887	616	12	are	be	AUX
ejpam-5887	616	13	n.	n.	NOUN
ejpam-5887	616	14	m.	m.	NOUN
ejpam-5887	616	15	noureldeen	noureldeen	NOUN
ejpam-5887	616	16	et	et	PROPN
ejpam-5887	616	17	al	al	PROPN
ejpam-5887	616	18	.	.	PUNCT
ejpam-5887	616	19	/	/	SYM
ejpam-5887	616	20	eur	eur	PROPN
ejpam-5887	616	21	.	.	PUNCT
ejpam-5887	617	1	j.	j.	PROPN
ejpam-5887	617	2	pure	pure	PROPN
ejpam-5887	617	3	appl	appl	PROPN
ejpam-5887	617	4	.	.	PROPN
ejpam-5887	617	5	math	math	PROPN
ejpam-5887	617	6	,	,	PUNCT
ejpam-5887	617	7	18	18	NUM
ejpam-5887	617	8	(	(	PUNCT
ejpam-5887	617	9	2	2	NUM
ejpam-5887	617	10	)	)	PUNCT
ejpam-5887	617	11	(	(	PUNCT
ejpam-5887	617	12	2025	2025	NUM
ejpam-5887	617	13	)	)	PUNCT
ejpam-5887	617	14	,	,	PUNCT
ejpam-5887	617	15	5887	5887	NUM
ejpam-5887	617	16	17	17	NUM
ejpam-5887	617	17	of	of	ADP
ejpam-5887	617	18	21	21	NUM
ejpam-5887	617	19	isomorphic	isomorphic	ADJ
ejpam-5887	617	20	.	.	PUNCT
ejpam-5887	618	1	hence	hence	ADV
ejpam-5887	618	2	,	,	PUNCT
ejpam-5887	618	3	we	we	PRON
ejpam-5887	618	4	use	use	VERB
ejpam-5887	618	5	the	the	DET
ejpam-5887	618	6	notation	notation	NOUN
ejpam-5887	618	7	p	p	PROPN
ejpam-5887	618	8	(	(	PUNCT
ejpam-5887	618	9	n.cm	n.cm	PROPN
ejpam-5887	618	10	)	)	PUNCT
ejpam-5887	618	11	.	.	PUNCT
ejpam-5887	619	1	in	in	ADP
ejpam-5887	619	2	order	order	NOUN
ejpam-5887	619	3	to	to	PART
ejpam-5887	619	4	establish	establish	VERB
ejpam-5887	619	5	the	the	DET
ejpam-5887	619	6	necessary	necessary	ADJ
ejpam-5887	619	7	condition	condition	NOUN
ejpam-5887	619	8	for	for	ADP
ejpam-5887	619	9	the	the	DET
ejpam-5887	619	10	path	path	NOUN
ejpam-5887	619	11	union	union	NOUN
ejpam-5887	619	12	of	of	ADP
ejpam-5887	619	13	a	a	DET
ejpam-5887	619	14	cycle	cycle	NOUN
ejpam-5887	619	15	graph	graph	NOUN
ejpam-5887	619	16	p	p	X
ejpam-5887	619	17	(	(	PUNCT
ejpam-5887	619	18	n.cm	n.cm	PROPN
ejpam-5887	619	19	)	)	PUNCT
ejpam-5887	619	20	to	to	PART
ejpam-5887	619	21	admit	admit	VERB
ejpam-5887	619	22	an	an	DET
ejpam-5887	619	23	edge	edge	NOUN
ejpam-5887	619	24	k	k	NOUN
ejpam-5887	619	25	-	-	PUNCT
ejpam-5887	619	26	product	product	NOUN
ejpam-5887	619	27	cordial	cordial	ADJ
ejpam-5887	619	28	labeling	labeling	NOUN
ejpam-5887	619	29	,	,	PUNCT
ejpam-5887	619	30	we	we	PRON
ejpam-5887	619	31	prove	prove	VERB
ejpam-5887	619	32	the	the	DET
ejpam-5887	619	33	following	follow	VERB
ejpam-5887	619	34	general	general	ADJ
ejpam-5887	619	35	result	result	NOUN
ejpam-5887	619	36	.	.	PUNCT
ejpam-5887	620	1	theorem	theorem	NOUN
ejpam-5887	620	2	19	19	NUM
ejpam-5887	620	3	.	.	PUNCT
ejpam-5887	621	1	any	any	DET
ejpam-5887	621	2	graph	graph	NOUN
ejpam-5887	621	3	g	g	NOUN
ejpam-5887	621	4	with	with	ADP
ejpam-5887	621	5	⌊	⌊	PROPN
ejpam-5887	621	6	|v	|v	VERB
ejpam-5887	621	7	|	|	ADV
ejpam-5887	621	8	k	k	PROPN
ejpam-5887	621	9	⌋	⌋	NOUN
ejpam-5887	621	10	<	<	X
ejpam-5887	621	11	⌊	⌊	PROPN
ejpam-5887	621	12	|e|	|e|	DET
ejpam-5887	621	13	k	k	PROPN
ejpam-5887	621	14	⌋	⌋	PROPN
ejpam-5887	621	15	does	do	AUX
ejpam-5887	621	16	not	not	PART
ejpam-5887	621	17	an	an	DET
ejpam-5887	621	18	admit	admit	NOUN
ejpam-5887	621	19	edge	edge	NOUN
ejpam-5887	621	20	k	k	NOUN
ejpam-5887	621	21	-	-	PUNCT
ejpam-5887	621	22	product	product	NOUN
ejpam-5887	621	23	cordial	cordial	ADJ
ejpam-5887	621	24	labeling	labeling	NOUN
ejpam-5887	621	25	.	.	PUNCT
ejpam-5887	622	1	proof	proof	NOUN
ejpam-5887	622	2	.	.	PUNCT
ejpam-5887	623	1	let	let	VERB
ejpam-5887	623	2	f	f	PRON
ejpam-5887	623	3	be	be	AUX
ejpam-5887	623	4	an	an	DET
ejpam-5887	623	5	edge	edge	NOUN
ejpam-5887	623	6	k	k	NOUN
ejpam-5887	623	7	-	-	PUNCT
ejpam-5887	623	8	product	product	NOUN
ejpam-5887	623	9	cordial	cordial	ADJ
ejpam-5887	623	10	labeling	labeling	NOUN
ejpam-5887	623	11	of	of	ADP
ejpam-5887	623	12	a	a	DET
ejpam-5887	623	13	graph	graph	NOUN
ejpam-5887	623	14	g	g	NOUN
ejpam-5887	623	15	with	with	ADP
ejpam-5887	623	16	⌊	⌊	PROPN
ejpam-5887	623	17	|v	|v	VERB
ejpam-5887	623	18	|	|	ADV
ejpam-5887	623	19	k	k	PROPN
ejpam-5887	623	20	⌋	⌋	NOUN
ejpam-5887	623	21	<	<	X
ejpam-5887	623	22	⌊	⌊	PROPN
ejpam-5887	623	23	|e|	|e|	DET
ejpam-5887	623	24	k	k	PROPN
ejpam-5887	623	25	⌋.	⌋.	ADV
ejpam-5887	623	26	then	then	ADV
ejpam-5887	623	27	ef	ef	PROPN
ejpam-5887	623	28	(	(	PUNCT
ejpam-5887	623	29	i	i	NOUN
ejpam-5887	623	30	)	)	PUNCT
ejpam-5887	623	31	is	be	AUX
ejpam-5887	623	32	either	either	CCONJ
ejpam-5887	623	33	⌊	⌊	PROPN
ejpam-5887	623	34	|e|	|e|	DET
ejpam-5887	623	35	k	k	PROPN
ejpam-5887	623	36	⌋	⌋	NOUN
ejpam-5887	623	37	or	or	CCONJ
ejpam-5887	623	38	⌊	⌊	VERB
ejpam-5887	623	39	|e|	|e|	DET
ejpam-5887	623	40	k	k	PROPN
ejpam-5887	623	41	⌋+1	⌋+1	PROPN
ejpam-5887	623	42	and	and	CCONJ
ejpam-5887	623	43	vf∗(i	vf∗(i	ADJ
ejpam-5887	623	44	)	)	PUNCT
ejpam-5887	623	45	is	be	AUX
ejpam-5887	623	46	either	either	CCONJ
ejpam-5887	623	47	⌊	⌊	AUX
ejpam-5887	623	48	|v	|v	VERB
ejpam-5887	623	49	|	|	ADV
ejpam-5887	623	50	k	k	PROPN
ejpam-5887	623	51	⌋	⌋	NOUN
ejpam-5887	623	52	or	or	CCONJ
ejpam-5887	623	53	⌊	⌊	AUX
ejpam-5887	623	54	|v	|v	VERB
ejpam-5887	623	55	|	|	ADV
ejpam-5887	623	56	k	k	PROPN
ejpam-5887	623	57	⌋+1	⌋+1	PROPN
ejpam-5887	623	58	(	(	PUNCT
ejpam-5887	623	59	i	i	NOUN
ejpam-5887	623	60	=	=	NOUN
ejpam-5887	623	61	0	0	NUM
ejpam-5887	623	62	,	,	PUNCT
ejpam-5887	623	63	1	1	NUM
ejpam-5887	623	64	,	,	PUNCT
ejpam-5887	623	65	...	...	PUNCT
ejpam-5887	623	66	,	,	PUNCT
ejpam-5887	623	67	k−1	k−1	PROPN
ejpam-5887	623	68	)	)	PUNCT
ejpam-5887	623	69	.	.	PUNCT
ejpam-5887	624	1	if	if	SCONJ
ejpam-5887	624	2	ef	ef	PROPN
ejpam-5887	624	3	(	(	PUNCT
ejpam-5887	624	4	0	0	NUM
ejpam-5887	624	5	)	)	PUNCT
ejpam-5887	624	6	=	=	PUNCT
ejpam-5887	624	7	⌊	⌊	VERB
ejpam-5887	624	8	|e|	|e|	DET
ejpam-5887	624	9	k	k	PROPN
ejpam-5887	624	10	⌋	⌋	PROPN
ejpam-5887	624	11	,	,	PUNCT
ejpam-5887	624	12	then	then	ADV
ejpam-5887	624	13	vf∗(0	vf∗(0	VERB
ejpam-5887	624	14	)	)	PUNCT
ejpam-5887	624	15	≥	≥	NOUN
ejpam-5887	624	16	⌊	⌊	PROPN
ejpam-5887	624	17	|e|	|e|	DET
ejpam-5887	624	18	k	k	X
ejpam-5887	624	19	⌋+	⌋+	ADJ
ejpam-5887	624	20	1	1	X
ejpam-5887	624	21	>	>	X
ejpam-5887	624	22	⌊	⌊	X
ejpam-5887	624	23	|v	|v	ADV
ejpam-5887	625	1	|	|	ADV
ejpam-5887	625	2	k	k	PROPN
ejpam-5887	625	3	⌋+	⌋+	ADJ
ejpam-5887	625	4	1	1	NUM
ejpam-5887	625	5	,	,	PUNCT
ejpam-5887	625	6	which	which	PRON
ejpam-5887	625	7	is	be	AUX
ejpam-5887	625	8	a	a	DET
ejpam-5887	625	9	contradiction	contradiction	NOUN
ejpam-5887	625	10	.	.	PUNCT
ejpam-5887	626	1	therefore	therefore	ADV
ejpam-5887	626	2	,	,	PUNCT
ejpam-5887	626	3	ef	ef	X
ejpam-5887	626	4	(	(	PUNCT
ejpam-5887	626	5	0	0	NUM
ejpam-5887	626	6	)	)	PUNCT
ejpam-5887	626	7	=	=	PUNCT
ejpam-5887	626	8	⌊	⌊	VERB
ejpam-5887	626	9	|e|	|e|	DET
ejpam-5887	626	10	k	k	PROPN
ejpam-5887	626	11	⌋	⌋	PROPN
ejpam-5887	627	1	+	+	CCONJ
ejpam-5887	627	2	1	1	NUM
ejpam-5887	627	3	,	,	PUNCT
ejpam-5887	627	4	which	which	PRON
ejpam-5887	627	5	results	result	VERB
ejpam-5887	627	6	vf∗(0	vf∗(0	NOUN
ejpam-5887	627	7	)	)	PUNCT
ejpam-5887	627	8	≥	≥	NOUN
ejpam-5887	628	1	⌊	⌊	VERB
ejpam-5887	628	2	|e|	|e|	DET
ejpam-5887	628	3	k	k	PROPN
ejpam-5887	628	4	⌋	⌋	PROPN
ejpam-5887	629	1	+	+	CCONJ
ejpam-5887	629	2	2	2	NUM
ejpam-5887	629	3	>	>	PUNCT
ejpam-5887	629	4	⌊	⌊	X
ejpam-5887	629	5	|v	|v	ADP
ejpam-5887	630	1	|	|	ADV
ejpam-5887	630	2	k	k	PROPN
ejpam-5887	630	3	⌋	⌋	NOUN
ejpam-5887	630	4	+	+	CCONJ
ejpam-5887	630	5	1	1	NUM
ejpam-5887	630	6	,	,	PUNCT
ejpam-5887	630	7	a	a	DET
ejpam-5887	630	8	contradiction	contradiction	NOUN
ejpam-5887	630	9	again	again	ADV
ejpam-5887	630	10	.	.	PUNCT
ejpam-5887	631	1	hence	hence	ADV
ejpam-5887	631	2	,	,	PUNCT
ejpam-5887	631	3	g	g	PROPN
ejpam-5887	631	4	is	be	AUX
ejpam-5887	631	5	not	not	PART
ejpam-5887	631	6	an	an	DET
ejpam-5887	631	7	edge	edge	NOUN
ejpam-5887	631	8	k	k	NOUN
ejpam-5887	631	9	-	-	PUNCT
ejpam-5887	631	10	product	product	NOUN
ejpam-5887	631	11	cordial	cordial	ADJ
ejpam-5887	631	12	graph	graph	NOUN
ejpam-5887	631	13	.	.	PUNCT
ejpam-5887	632	1	theorem	theorem	NOUN
ejpam-5887	632	2	20	20	NUM
ejpam-5887	632	3	.	.	PUNCT
ejpam-5887	633	1	the	the	DET
ejpam-5887	633	2	path	path	PROPN
ejpam-5887	633	3	union	union	PROPN
ejpam-5887	633	4	of	of	ADP
ejpam-5887	633	5	cycle	cycle	NOUN
ejpam-5887	633	6	graph	graph	NOUN
ejpam-5887	633	7	p	p	X
ejpam-5887	633	8	(	(	PUNCT
ejpam-5887	633	9	n.cm	n.cm	PROPN
ejpam-5887	633	10	)	)	PUNCT
ejpam-5887	633	11	does	do	AUX
ejpam-5887	633	12	not	not	PART
ejpam-5887	633	13	admit	admit	VERB
ejpam-5887	633	14	an	an	DET
ejpam-5887	633	15	edge	edge	NOUN
ejpam-5887	633	16	k	k	NOUN
ejpam-5887	633	17	-	-	PUNCT
ejpam-5887	633	18	product	product	NOUN
ejpam-5887	633	19	cordial	cordial	ADJ
ejpam-5887	633	20	labeling	labeling	NOUN
ejpam-5887	633	21	if	if	SCONJ
ejpam-5887	633	22	n	n	NUM
ejpam-5887	633	23	≥	≥	NOUN
ejpam-5887	633	24	k.	k.	PROPN
ejpam-5887	633	25	proof	proof	PROPN
ejpam-5887	633	26	.	.	PUNCT
ejpam-5887	634	1	for	for	ADP
ejpam-5887	634	2	the	the	DET
ejpam-5887	634	3	path	path	NOUN
ejpam-5887	634	4	union	union	PROPN
ejpam-5887	634	5	of	of	ADP
ejpam-5887	634	6	cycle	cycle	NOUN
ejpam-5887	634	7	graph	graph	NOUN
ejpam-5887	634	8	p	p	X
ejpam-5887	634	9	(	(	PUNCT
ejpam-5887	634	10	n.cm	n.cm	PROPN
ejpam-5887	634	11	)	)	PUNCT
ejpam-5887	634	12	,	,	PUNCT
ejpam-5887	634	13	we	we	PRON
ejpam-5887	634	14	have	have	AUX
ejpam-5887	634	15	|v	|v	VERB
ejpam-5887	634	16	|	|	ADV
ejpam-5887	634	17	=	=	SYM
ejpam-5887	634	18	nm	nm	NOUN
ejpam-5887	634	19	and	and	CCONJ
ejpam-5887	634	20	|e|	|e|	NOUN
ejpam-5887	634	21	=	=	PUNCT
ejpam-5887	634	22	nm	nm	NOUN
ejpam-5887	634	23	+	+	CCONJ
ejpam-5887	634	24	n	n	CCONJ
ejpam-5887	634	25	−	−	PROPN
ejpam-5887	634	26	1	1	NUM
ejpam-5887	634	27	.	.	PUNCT
ejpam-5887	635	1	let	let	VERB
ejpam-5887	635	2	f	f	PRON
ejpam-5887	635	3	be	be	AUX
ejpam-5887	635	4	an	an	DET
ejpam-5887	635	5	edge	edge	NOUN
ejpam-5887	635	6	k	k	NOUN
ejpam-5887	635	7	-	-	PUNCT
ejpam-5887	635	8	product	product	NOUN
ejpam-5887	635	9	cordial	cordial	ADJ
ejpam-5887	635	10	labeling	labeling	NOUN
ejpam-5887	635	11	of	of	ADP
ejpam-5887	635	12	p	p	PROPN
ejpam-5887	635	13	(	(	PUNCT
ejpam-5887	635	14	n.cm	n.cm	PROPN
ejpam-5887	635	15	)	)	PUNCT
ejpam-5887	635	16	;	;	PUNCT
ejpam-5887	636	1	n	n	DET
ejpam-5887	636	2	≥	≥	NOUN
ejpam-5887	636	3	k.	k.	X
ejpam-5887	637	1	we	we	PRON
ejpam-5887	637	2	have	have	VERB
ejpam-5887	637	3	the	the	DET
ejpam-5887	637	4	following	follow	VERB
ejpam-5887	637	5	two	two	NUM
ejpam-5887	637	6	cases	case	NOUN
ejpam-5887	637	7	.	.	PUNCT
ejpam-5887	638	1	case(i	case(i	NOUN
ejpam-5887	638	2	):	):	PUNCT
ejpam-5887	638	3	for	for	ADP
ejpam-5887	638	4	n	n	NOUN
ejpam-5887	638	5	=	=	SYM
ejpam-5887	638	6	k	k	NOUN
ejpam-5887	638	7	,	,	PUNCT
ejpam-5887	638	8	we	we	PRON
ejpam-5887	638	9	have	have	VERB
ejpam-5887	638	10	ef	ef	X
ejpam-5887	638	11	(	(	PUNCT
ejpam-5887	638	12	i	i	NOUN
ejpam-5887	638	13	)	)	PUNCT
ejpam-5887	638	14	=	=	VERB
ejpam-5887	638	15	m	m	ADJ
ejpam-5887	638	16	or	or	CCONJ
ejpam-5887	638	17	m+1	m+1	PRON
ejpam-5887	638	18	(	(	PUNCT
ejpam-5887	638	19	i	i	NOUN
ejpam-5887	638	20	=	=	NOUN
ejpam-5887	638	21	0	0	NUM
ejpam-5887	638	22	,	,	PUNCT
ejpam-5887	638	23	1	1	NUM
ejpam-5887	638	24	,	,	PUNCT
ejpam-5887	638	25	2	2	NUM
ejpam-5887	638	26	,	,	PUNCT
ejpam-5887	638	27	...	...	PUNCT
ejpam-5887	638	28	,	,	PUNCT
ejpam-5887	638	29	k−1	k−1	PROPN
ejpam-5887	638	30	)	)	PUNCT
ejpam-5887	638	31	and	and	CCONJ
ejpam-5887	638	32	vf∗(i	vf∗(i	ADJ
ejpam-5887	638	33	)	)	PUNCT
ejpam-5887	638	34	=	=	SYM
ejpam-5887	639	1	m	m	VERB
ejpam-5887	639	2	(	(	PUNCT
ejpam-5887	639	3	i	i	NOUN
ejpam-5887	639	4	=	=	NOUN
ejpam-5887	639	5	0	0	NUM
ejpam-5887	639	6	,	,	PUNCT
ejpam-5887	639	7	1	1	NUM
ejpam-5887	639	8	,	,	PUNCT
ejpam-5887	639	9	2	2	NUM
ejpam-5887	639	10	,	,	PUNCT
ejpam-5887	639	11	...	...	PUNCT
ejpam-5887	639	12	,	,	PUNCT
ejpam-5887	639	13	k	k	PROPN
ejpam-5887	640	1	−	−	PROPN
ejpam-5887	640	2	1	1	NUM
ejpam-5887	640	3	)	)	PUNCT
ejpam-5887	640	4	.	.	PUNCT
ejpam-5887	641	1	if	if	SCONJ
ejpam-5887	641	2	ef	ef	PROPN
ejpam-5887	641	3	(	(	PUNCT
ejpam-5887	641	4	0	0	NUM
ejpam-5887	641	5	)	)	PUNCT
ejpam-5887	641	6	=	=	SYM
ejpam-5887	641	7	m	m	PROPN
ejpam-5887	641	8	,	,	PUNCT
ejpam-5887	641	9	then	then	ADV
ejpam-5887	641	10	vf∗(0	vf∗(0	VERB
ejpam-5887	641	11	)	)	PUNCT
ejpam-5887	641	12	≥	≥	NOUN
ejpam-5887	641	13	m+	m+	NUM
ejpam-5887	641	14	1	1	NUM
ejpam-5887	641	15	,	,	PUNCT
ejpam-5887	641	16	which	which	PRON
ejpam-5887	641	17	is	be	AUX
ejpam-5887	641	18	a	a	DET
ejpam-5887	641	19	contradiction	contradiction	NOUN
ejpam-5887	641	20	.	.	PUNCT
ejpam-5887	642	1	case	case	NOUN
ejpam-5887	642	2	(	(	PUNCT
ejpam-5887	642	3	ii	ii	NOUN
ejpam-5887	642	4	):	):	PUNCT
ejpam-5887	642	5	for	for	ADP
ejpam-5887	642	6	n	n	PRON
ejpam-5887	642	7	≥	≥	NOUN
ejpam-5887	642	8	k	k	NOUN
ejpam-5887	643	1	+	+	CCONJ
ejpam-5887	643	2	1	1	NUM
ejpam-5887	643	3	,	,	PUNCT
ejpam-5887	643	4	we	we	PRON
ejpam-5887	643	5	have	have	VERB
ejpam-5887	643	6	⌊nm+n−1	⌊nm+n−1	ADJ
ejpam-5887	643	7	k	k	X
ejpam-5887	643	8	⌋	⌋	PROPN
ejpam-5887	643	9	>	>	X
ejpam-5887	643	10	⌊nmk	⌊nmk	PROPN
ejpam-5887	643	11	⌋.	⌋.	X
ejpam-5887	643	12	by	by	ADP
ejpam-5887	643	13	theorem	theorem	NOUN
ejpam-5887	643	14	19	19	NUM
ejpam-5887	643	15	,	,	PUNCT
ejpam-5887	643	16	p	p	X
ejpam-5887	643	17	(	(	PUNCT
ejpam-5887	643	18	n.cm	n.cm	ADJ
ejpam-5887	643	19	)	)	PUNCT
ejpam-5887	643	20	is	be	AUX
ejpam-5887	643	21	not	not	PART
ejpam-5887	643	22	an	an	DET
ejpam-5887	643	23	edge	edge	NOUN
ejpam-5887	643	24	k	k	NOUN
ejpam-5887	643	25	-	-	PUNCT
ejpam-5887	643	26	product	product	NOUN
ejpam-5887	643	27	cordial	cordial	ADJ
ejpam-5887	643	28	graph	graph	NOUN
ejpam-5887	643	29	.	.	PUNCT
ejpam-5887	644	1	hence	hence	ADV
ejpam-5887	644	2	,	,	PUNCT
ejpam-5887	644	3	p	p	X
ejpam-5887	644	4	(	(	PUNCT
ejpam-5887	644	5	n.cm	n.cm	ADJ
ejpam-5887	644	6	)	)	PUNCT
ejpam-5887	644	7	is	be	AUX
ejpam-5887	644	8	not	not	PART
ejpam-5887	644	9	an	an	DET
ejpam-5887	644	10	edge	edge	NOUN
ejpam-5887	644	11	k	k	NOUN
ejpam-5887	644	12	-	-	PUNCT
ejpam-5887	644	13	product	product	NOUN
ejpam-5887	644	14	cordial	cordial	ADJ
ejpam-5887	644	15	graph	graph	NOUN
ejpam-5887	644	16	if	if	SCONJ
ejpam-5887	644	17	n	n	NUM
ejpam-5887	644	18	≥	≥	PROPN
ejpam-5887	644	19	k.	k.	PROPN
ejpam-5887	644	20	theorem	theorem	PROPN
ejpam-5887	644	21	21	21	NUM
ejpam-5887	644	22	.	.	PUNCT
ejpam-5887	645	1	the	the	DET
ejpam-5887	645	2	path	path	PROPN
ejpam-5887	645	3	union	union	PROPN
ejpam-5887	645	4	of	of	ADP
ejpam-5887	645	5	cycle	cycle	NOUN
ejpam-5887	645	6	graph	graph	NOUN
ejpam-5887	645	7	p	p	X
ejpam-5887	645	8	(	(	PUNCT
ejpam-5887	645	9	n.cm	n.cm	PROPN
ejpam-5887	645	10	)	)	PUNCT
ejpam-5887	645	11	does	do	AUX
ejpam-5887	645	12	not	not	PART
ejpam-5887	645	13	admit	admit	VERB
ejpam-5887	645	14	an	an	DET
ejpam-5887	645	15	edge	edge	NOUN
ejpam-5887	645	16	k	k	NOUN
ejpam-5887	645	17	-	-	PUNCT
ejpam-5887	645	18	product	product	NOUN
ejpam-5887	645	19	cordial	cordial	ADJ
ejpam-5887	645	20	labeling	labeling	NOUN
ejpam-5887	645	21	if	if	SCONJ
ejpam-5887	645	22	m	m	NOUN
ejpam-5887	645	23	is	be	AUX
ejpam-5887	645	24	a	a	DET
ejpam-5887	645	25	multiple	multiple	NOUN
ejpam-5887	645	26	of	of	ADP
ejpam-5887	645	27	k.	k.	NOUN
ejpam-5887	645	28	proof	proof	PROPN
ejpam-5887	645	29	.	.	PUNCT
ejpam-5887	646	1	let	let	VERB
ejpam-5887	646	2	m	m	VERB
ejpam-5887	646	3	=	=	VERB
ejpam-5887	646	4	kt	kt	PROPN
ejpam-5887	646	5	;	;	PUNCT
ejpam-5887	646	6	t	t	PROPN
ejpam-5887	646	7	≥	≥	NUM
ejpam-5887	646	8	1	1	NUM
ejpam-5887	646	9	.	.	PUNCT
ejpam-5887	647	1	for	for	ADP
ejpam-5887	647	2	the	the	DET
ejpam-5887	647	3	path	path	NOUN
ejpam-5887	647	4	union	union	PROPN
ejpam-5887	647	5	of	of	ADP
ejpam-5887	647	6	cycle	cycle	NOUN
ejpam-5887	647	7	graph	graph	NOUN
ejpam-5887	647	8	p	p	PROPN
ejpam-5887	647	9	(	(	PUNCT
ejpam-5887	647	10	n.ckt	n.ckt	PROPN
ejpam-5887	647	11	)	)	PUNCT
ejpam-5887	647	12	,	,	PUNCT
ejpam-5887	647	13	we	we	PRON
ejpam-5887	647	14	have	have	VERB
ejpam-5887	647	15	|v	|v	VERB
ejpam-5887	647	16	|	|	ADV
ejpam-5887	647	17	=	=	SYM
ejpam-5887	647	18	ktn	ktn	PROPN
ejpam-5887	647	19	and	and	CCONJ
ejpam-5887	647	20	|e|	|e|	PROPN
ejpam-5887	647	21	=	=	PROPN
ejpam-5887	647	22	ktn+n−1	ktn+n−1	PROPN
ejpam-5887	647	23	.	.	PUNCT
ejpam-5887	648	1	let	let	VERB
ejpam-5887	648	2	f	f	PRON
ejpam-5887	648	3	be	be	AUX
ejpam-5887	648	4	an	an	DET
ejpam-5887	648	5	edge	edge	NOUN
ejpam-5887	648	6	k	k	NOUN
ejpam-5887	648	7	-	-	PUNCT
ejpam-5887	648	8	product	product	NOUN
ejpam-5887	648	9	cordial	cordial	ADJ
ejpam-5887	648	10	labeling	labeling	NOUN
ejpam-5887	648	11	of	of	ADP
ejpam-5887	648	12	p	p	PROPN
ejpam-5887	648	13	(	(	PUNCT
ejpam-5887	648	14	n.ckt	n.ckt	PROPN
ejpam-5887	648	15	)	)	PUNCT
ejpam-5887	648	16	;	;	PUNCT
ejpam-5887	648	17	n	n	CCONJ
ejpam-5887	648	18	<	<	X
ejpam-5887	648	19	k.	k.	PROPN
ejpam-5887	648	20	since	since	SCONJ
ejpam-5887	648	21	n−1	n−1	PROPN
ejpam-5887	648	22	<	<	X
ejpam-5887	648	23	k	k	PROPN
ejpam-5887	648	24	,	,	PUNCT
ejpam-5887	648	25	we	we	PRON
ejpam-5887	648	26	have	have	VERB
ejpam-5887	648	27	ef	ef	X
ejpam-5887	648	28	(	(	PUNCT
ejpam-5887	648	29	i	i	NOUN
ejpam-5887	648	30	)	)	PUNCT
ejpam-5887	648	31	=	=	SYM
ejpam-5887	648	32	tn	tn	PROPN
ejpam-5887	648	33	or	or	CCONJ
ejpam-5887	648	34	tn+1	tn+1	NUM
ejpam-5887	648	35	and	and	CCONJ
ejpam-5887	648	36	vf∗(i	vf∗(i	ADJ
ejpam-5887	648	37	)	)	PUNCT
ejpam-5887	649	1	=	=	SYM
ejpam-5887	649	2	tn	tn	PROPN
ejpam-5887	649	3	(	(	PUNCT
ejpam-5887	649	4	i	i	NOUN
ejpam-5887	649	5	=	=	NOUN
ejpam-5887	649	6	0	0	NUM
ejpam-5887	649	7	,	,	PUNCT
ejpam-5887	649	8	1	1	NUM
ejpam-5887	649	9	,	,	PUNCT
ejpam-5887	649	10	2	2	NUM
ejpam-5887	649	11	,	,	PUNCT
ejpam-5887	649	12	...	...	PUNCT
ejpam-5887	649	13	,	,	PUNCT
ejpam-5887	649	14	k−1	k−1	PROPN
ejpam-5887	649	15	)	)	PUNCT
ejpam-5887	649	16	.	.	PUNCT
ejpam-5887	650	1	if	if	SCONJ
ejpam-5887	650	2	ef	ef	PROPN
ejpam-5887	650	3	(	(	PUNCT
ejpam-5887	650	4	0	0	NUM
ejpam-5887	650	5	)	)	PUNCT
ejpam-5887	650	6	=	=	SYM
ejpam-5887	650	7	tn	tn	PROPN
ejpam-5887	650	8	,	,	PUNCT
ejpam-5887	650	9	then	then	ADV
ejpam-5887	650	10	vf∗(0	vf∗(0	VERB
ejpam-5887	650	11	)	)	PUNCT
ejpam-5887	650	12	≥	≥	NUM
ejpam-5887	650	13	tn+1	tn+1	PROPN
ejpam-5887	650	14	.	.	PUNCT
ejpam-5887	651	1	therefore	therefore	ADV
ejpam-5887	651	2	,	,	PUNCT
ejpam-5887	651	3	|vf∗(0)−vf∗(i)|	|vf∗(0)−vf∗(i)|	PRON
ejpam-5887	651	4	≥	≥	NOUN
ejpam-5887	651	5	2	2	NUM
ejpam-5887	651	6	for	for	ADP
ejpam-5887	651	7	some	some	DET
ejpam-5887	651	8	i	i	PRON
ejpam-5887	651	9	∈	∈	PROPN
ejpam-5887	651	10	{	{	PUNCT
ejpam-5887	651	11	1	1	NUM
ejpam-5887	651	12	,	,	PUNCT
ejpam-5887	651	13	2	2	NUM
ejpam-5887	651	14	,	,	PUNCT
ejpam-5887	651	15	...	...	PUNCT
ejpam-5887	651	16	,	,	PUNCT
ejpam-5887	651	17	k−1	k−1	PROPN
ejpam-5887	651	18	}	}	PUNCT
ejpam-5887	651	19	,	,	PUNCT
ejpam-5887	651	20	which	which	PRON
ejpam-5887	651	21	is	be	AUX
ejpam-5887	651	22	a	a	DET
ejpam-5887	651	23	contradiction	contradiction	NOUN
ejpam-5887	651	24	.	.	PUNCT
ejpam-5887	652	1	thus	thus	ADV
ejpam-5887	652	2	,	,	PUNCT
ejpam-5887	652	3	f	f	PROPN
ejpam-5887	652	4	is	be	AUX
ejpam-5887	652	5	not	not	PART
ejpam-5887	652	6	an	an	DET
ejpam-5887	652	7	edge	edge	NOUN
ejpam-5887	652	8	k	k	NOUN
ejpam-5887	652	9	-	-	PUNCT
ejpam-5887	652	10	product	product	NOUN
ejpam-5887	652	11	cordial	cordial	ADJ
ejpam-5887	652	12	labeling	labeling	NOUN
ejpam-5887	652	13	of	of	ADP
ejpam-5887	652	14	p	p	PROPN
ejpam-5887	652	15	(	(	PUNCT
ejpam-5887	652	16	n.ckt	n.ckt	PROPN
ejpam-5887	652	17	)	)	PUNCT
ejpam-5887	652	18	;	;	PUNCT
ejpam-5887	652	19	n	n	CCONJ
ejpam-5887	652	20	<	<	X
ejpam-5887	652	21	k.	k.	X
ejpam-5887	652	22	by	by	ADP
ejpam-5887	652	23	theorem	theorem	NOUN
ejpam-5887	652	24	20	20	NUM
ejpam-5887	652	25	,	,	PUNCT
ejpam-5887	652	26	p	p	X
ejpam-5887	652	27	(	(	PUNCT
ejpam-5887	652	28	n.cm	n.cm	PROPN
ejpam-5887	652	29	)	)	PUNCT
ejpam-5887	652	30	;	;	PUNCT
ejpam-5887	652	31	n	n	DET
ejpam-5887	652	32	≥	≥	NOUN
ejpam-5887	652	33	k	k	X
ejpam-5887	652	34	is	be	AUX
ejpam-5887	652	35	not	not	PART
ejpam-5887	652	36	an	an	DET
ejpam-5887	652	37	edge	edge	NOUN
ejpam-5887	652	38	k	k	NOUN
ejpam-5887	652	39	-	-	PUNCT
ejpam-5887	652	40	product	product	NOUN
ejpam-5887	652	41	cordial	cordial	ADJ
ejpam-5887	652	42	graph	graph	NOUN
ejpam-5887	652	43	.	.	PUNCT
ejpam-5887	653	1	hence	hence	ADV
ejpam-5887	653	2	,	,	PUNCT
ejpam-5887	653	3	p	p	X
ejpam-5887	653	4	(	(	PUNCT
ejpam-5887	653	5	n.ckt	n.ckt	NOUN
ejpam-5887	653	6	)	)	PUNCT
ejpam-5887	653	7	is	be	AUX
ejpam-5887	653	8	not	not	PART
ejpam-5887	653	9	an	an	DET
ejpam-5887	653	10	edge	edge	NOUN
ejpam-5887	653	11	k	k	NOUN
ejpam-5887	653	12	-	-	PUNCT
ejpam-5887	653	13	product	product	NOUN
ejpam-5887	653	14	cordial	cordial	ADJ
ejpam-5887	653	15	graph	graph	NOUN
ejpam-5887	653	16	.	.	PUNCT
ejpam-5887	654	1	theorem	theorem	NOUN
ejpam-5887	654	2	22	22	NUM
ejpam-5887	654	3	.	.	PUNCT
ejpam-5887	655	1	the	the	DET
ejpam-5887	655	2	path	path	PROPN
ejpam-5887	655	3	union	union	PROPN
ejpam-5887	655	4	of	of	ADP
ejpam-5887	655	5	cycle	cycle	NOUN
ejpam-5887	655	6	graph	graph	NOUN
ejpam-5887	655	7	p	p	X
ejpam-5887	655	8	(	(	PUNCT
ejpam-5887	655	9	n.cm	n.cm	PROPN
ejpam-5887	655	10	)	)	PUNCT
ejpam-5887	655	11	admits	admit	VERB
ejpam-5887	655	12	an	an	DET
ejpam-5887	655	13	edge	edge	NOUN
ejpam-5887	655	14	3	3	NUM
ejpam-5887	655	15	-	-	PUNCT
ejpam-5887	655	16	product	product	NOUN
ejpam-5887	655	17	cordial	cordial	ADJ
ejpam-5887	655	18	labeling	labeling	NOUN
ejpam-5887	655	19	if	if	SCONJ
ejpam-5887	655	20	and	and	CCONJ
ejpam-5887	655	21	only	only	ADV
ejpam-5887	655	22	if	if	SCONJ
ejpam-5887	655	23	n	n	PROPN
ejpam-5887	655	24	=	=	SYM
ejpam-5887	655	25	2	2	NUM
ejpam-5887	655	26	and	and	CCONJ
ejpam-5887	655	27	m	m	PROPN
ejpam-5887	655	28	≡	≡	PROPN
ejpam-5887	655	29	2	2	NUM
ejpam-5887	655	30	(	(	PUNCT
ejpam-5887	655	31	mod	mod	NOUN
ejpam-5887	655	32	3	3	NUM
ejpam-5887	655	33	)	)	PUNCT
ejpam-5887	655	34	.	.	PUNCT
ejpam-5887	656	1	proof	proof	NOUN
ejpam-5887	656	2	.	.	PUNCT
ejpam-5887	657	1	let	let	VERB
ejpam-5887	657	2	the	the	DET
ejpam-5887	657	3	vertex	vertex	NOUN
ejpam-5887	657	4	and	and	CCONJ
ejpam-5887	657	5	edge	edge	NOUN
ejpam-5887	657	6	set	set	NOUN
ejpam-5887	657	7	of	of	ADP
ejpam-5887	657	8	p	p	PROPN
ejpam-5887	657	9	(	(	PUNCT
ejpam-5887	657	10	n.cm	n.cm	PROPN
ejpam-5887	657	11	)	)	PUNCT
ejpam-5887	657	12	be	be	VERB
ejpam-5887	657	13	v	v	ADP
ejpam-5887	657	14	(	(	PUNCT
ejpam-5887	657	15	p	p	X
ejpam-5887	657	16	(	(	PUNCT
ejpam-5887	657	17	n.cm	n.cm	ADJ
ejpam-5887	657	18	)	)	PUNCT
ejpam-5887	657	19	)	)	PUNCT
ejpam-5887	658	1	=	=	PRON
ejpam-5887	658	2	{	{	PUNCT
ejpam-5887	658	3	vji	vji	NOUN
ejpam-5887	658	4	;	;	PUNCT
ejpam-5887	658	5	1	1	NUM
ejpam-5887	658	6	≤	≤	NUM
ejpam-5887	658	7	i	i	PRON
ejpam-5887	658	8	≤	≤	PROPN
ejpam-5887	658	9	n	n	CCONJ
ejpam-5887	658	10	,	,	PUNCT
ejpam-5887	658	11	1	1	NUM
ejpam-5887	658	12	≤	≤	NUM
ejpam-5887	658	13	j	j	PROPN
ejpam-5887	658	14	≤	≤	PROPN
ejpam-5887	658	15	m	m	PROPN
ejpam-5887	658	16	}	}	PUNCT
ejpam-5887	658	17	and	and	CCONJ
ejpam-5887	658	18	e(p	e(p	PROPN
ejpam-5887	658	19	(	(	PUNCT
ejpam-5887	658	20	n.cm	n.cm	ADJ
ejpam-5887	658	21	)	)	PUNCT
ejpam-5887	658	22	)	)	PUNCT
ejpam-5887	659	1	=	=	PRON
ejpam-5887	659	2	{	{	PUNCT
ejpam-5887	659	3	v1i	v1i	NUM
ejpam-5887	659	4	v1i+1	v1i+1	NOUN
ejpam-5887	659	5	,	,	PUNCT
ejpam-5887	659	6	v	v	NOUN
ejpam-5887	659	7	j	j	NOUN
ejpam-5887	660	1	i	i	PRON
ejpam-5887	660	2	v	v	VERB
ejpam-5887	660	3	j+1	j+1	ADV
ejpam-5887	661	1	i	i	PRON
ejpam-5887	661	2	,	,	PUNCT
ejpam-5887	661	3	vjnv	vjnv	NOUN
ejpam-5887	661	4	j+1	j+1	ADJ
ejpam-5887	661	5	n	n	NOUN
ejpam-5887	661	6	,	,	PUNCT
ejpam-5887	661	7	vmi	vmi	PROPN
ejpam-5887	661	8	v1i	v1i	PROPN
ejpam-5887	661	9	,	,	PUNCT
ejpam-5887	661	10	v	v	NOUN
ejpam-5887	661	11	m	m	PROPN
ejpam-5887	661	12	n	n	PRON
ejpam-5887	661	13	v1n	v1n	ADJ
ejpam-5887	661	14	;	;	PUNCT
ejpam-5887	661	15	1	1	NUM
ejpam-5887	661	16	≤	≤	NUM
ejpam-5887	661	17	i	i	PRON
ejpam-5887	661	18	≤	≤	ADJ
ejpam-5887	661	19	n	n	CCONJ
ejpam-5887	661	20	−	−	PROPN
ejpam-5887	661	21	1	1	NUM
ejpam-5887	661	22	,	,	PUNCT
ejpam-5887	661	23	1	1	NUM
ejpam-5887	661	24	≤	≤	NUM
ejpam-5887	661	25	j	j	PROPN
ejpam-5887	661	26	≤	≤	PROPN
ejpam-5887	661	27	m−	m−	PROPN
ejpam-5887	661	28	1	1	NUM
ejpam-5887	661	29	}	}	PUNCT
ejpam-5887	661	30	respectively	respectively	ADV
ejpam-5887	661	31	.	.	PUNCT
ejpam-5887	662	1	define	define	VERB
ejpam-5887	662	2	an	an	DET
ejpam-5887	662	3	edge	edge	NOUN
ejpam-5887	662	4	labeling	labeling	NOUN
ejpam-5887	662	5	f	f	NOUN
ejpam-5887	662	6	:	:	PUNCT
ejpam-5887	662	7	e(p	e(p	PROPN
ejpam-5887	662	8	(	(	PUNCT
ejpam-5887	662	9	2.c3t+2	2.c3t+2	NUM
ejpam-5887	662	10	)	)	PUNCT
ejpam-5887	662	11	)	)	PUNCT
ejpam-5887	663	1	→	→	PUNCT
ejpam-5887	663	2	{	{	PUNCT
ejpam-5887	663	3	0	0	NUM
ejpam-5887	663	4	,	,	PUNCT
ejpam-5887	663	5	1	1	NUM
ejpam-5887	663	6	,	,	PUNCT
ejpam-5887	663	7	2	2	NUM
ejpam-5887	663	8	}	}	PUNCT
ejpam-5887	663	9	for	for	ADP
ejpam-5887	663	10	t	t	PROPN
ejpam-5887	663	11	≥	≥	NUM
ejpam-5887	663	12	1	1	NUM
ejpam-5887	663	13	as	as	SCONJ
ejpam-5887	663	14	follows	follow	VERB
ejpam-5887	663	15	:	:	PUNCT
ejpam-5887	663	16	n.	n.	PROPN
ejpam-5887	663	17	m.	m.	NOUN
ejpam-5887	663	18	noureldeen	noureldeen	NOUN
ejpam-5887	663	19	et	et	PROPN
ejpam-5887	663	20	al	al	PROPN
ejpam-5887	663	21	.	.	PUNCT
ejpam-5887	663	22	/	/	SYM
ejpam-5887	663	23	eur	eur	PROPN
ejpam-5887	663	24	.	.	PUNCT
ejpam-5887	664	1	j.	j.	PROPN
ejpam-5887	664	2	pure	pure	PROPN
ejpam-5887	664	3	appl	appl	PROPN
ejpam-5887	664	4	.	.	PROPN
ejpam-5887	664	5	math	math	PROPN
ejpam-5887	664	6	,	,	PUNCT
ejpam-5887	664	7	18	18	NUM
ejpam-5887	664	8	(	(	PUNCT
ejpam-5887	664	9	2	2	NUM
ejpam-5887	664	10	)	)	PUNCT
ejpam-5887	664	11	(	(	PUNCT
ejpam-5887	664	12	2025	2025	NUM
ejpam-5887	664	13	)	)	PUNCT
ejpam-5887	664	14	,	,	PUNCT
ejpam-5887	664	15	5887	5887	NUM
ejpam-5887	664	16	18	18	NUM
ejpam-5887	664	17	of	of	ADP
ejpam-5887	664	18	21	21	NUM
ejpam-5887	664	19	f(v11v	f(v11v	SYM
ejpam-5887	664	20	1	1	NUM
ejpam-5887	664	21	2	2	NUM
ejpam-5887	664	22	)	)	PUNCT
ejpam-5887	664	23	=	=	SYM
ejpam-5887	664	24	0	0	NUM
ejpam-5887	664	25	,	,	PUNCT
ejpam-5887	664	26	f(v3t+2	f(v3t+2	PUNCT
ejpam-5887	664	27	i	i	PRON
ejpam-5887	664	28	v1i	v1i	VERB
ejpam-5887	664	29	)	)	PUNCT
ejpam-5887	665	1	=	=	SYM
ejpam-5887	665	2	1	1	NUM
ejpam-5887	665	3	;	;	PUNCT
ejpam-5887	665	4	1	1	NUM
ejpam-5887	665	5	≤	≤	NUM
ejpam-5887	665	6	i	i	X
ejpam-5887	665	7	≤	≤	NOUN
ejpam-5887	665	8	2	2	NUM
ejpam-5887	665	9	,	,	PUNCT
ejpam-5887	665	10	f(vj1v	f(vj1v	X
ejpam-5887	666	1	j+1	j+1	ADV
ejpam-5887	666	2	1	1	X
ejpam-5887	666	3	)	)	PUNCT
ejpam-5887	666	4	=	=	SYM
ejpam-5887	666	5	0	0	NUM
ejpam-5887	666	6	;	;	PUNCT
ejpam-5887	667	1	1	1	NUM
ejpam-5887	667	2	≤	≤	NUM
ejpam-5887	667	3	j	j	PROPN
ejpam-5887	667	4	≤	≤	PROPN
ejpam-5887	667	5	2	2	NUM
ejpam-5887	667	6	t	t	NOUN
ejpam-5887	667	7	,	,	PUNCT
ejpam-5887	667	8	f(v2t+j	f(v2t+j	PROPN
ejpam-5887	667	9	1	1	NUM
ejpam-5887	667	10	v2t+j+1	v2t+j+1	NUM
ejpam-5887	667	11	1	1	NUM
ejpam-5887	667	12	)	)	PUNCT
ejpam-5887	667	13	=	=	PRON
ejpam-5887	667	14	{	{	PUNCT
ejpam-5887	667	15	1	1	NUM
ejpam-5887	667	16	;	;	PUNCT
ejpam-5887	667	17	j	j	PROPN
ejpam-5887	667	18	≡	≡	PROPN
ejpam-5887	667	19	0	0	NUM
ejpam-5887	667	20	,	,	PUNCT
ejpam-5887	667	21	3	3	NUM
ejpam-5887	667	22	(	(	PUNCT
ejpam-5887	667	23	mod	mod	NOUN
ejpam-5887	667	24	4	4	NUM
ejpam-5887	667	25	)	)	PUNCT
ejpam-5887	667	26	2	2	NUM
ejpam-5887	667	27	;	;	PUNCT
ejpam-5887	667	28	j	j	PROPN
ejpam-5887	667	29	≡	≡	PROPN
ejpam-5887	667	30	1	1	NUM
ejpam-5887	667	31	,	,	PUNCT
ejpam-5887	667	32	2	2	NUM
ejpam-5887	667	33	(	(	PUNCT
ejpam-5887	667	34	mod	mod	NOUN
ejpam-5887	667	35	4	4	NUM
ejpam-5887	667	36	)	)	PUNCT
ejpam-5887	667	37	;	;	PUNCT
ejpam-5887	667	38	1	1	NUM
ejpam-5887	667	39	≤	≤	NUM
ejpam-5887	667	40	j	j	PROPN
ejpam-5887	667	41	≤	≤	PROPN
ejpam-5887	667	42	t+	t+	PUNCT
ejpam-5887	667	43	1	1	NUM
ejpam-5887	667	44	,	,	PUNCT
ejpam-5887	667	45	f(vj2v	f(vj2v	NOUN
ejpam-5887	667	46	j+1	j+1	ADJ
ejpam-5887	667	47	2	2	X
ejpam-5887	667	48	)	)	PUNCT
ejpam-5887	667	49	=	=	PRON
ejpam-5887	667	50	{	{	PUNCT
ejpam-5887	667	51	1	1	NUM
ejpam-5887	667	52	;	;	PUNCT
ejpam-5887	667	53	j	j	PROPN
ejpam-5887	667	54	≡	≡	PROPN
ejpam-5887	667	55	0	0	NUM
ejpam-5887	667	56	,	,	PUNCT
ejpam-5887	667	57	1	1	NUM
ejpam-5887	667	58	(	(	PUNCT
ejpam-5887	667	59	mod	mod	NOUN
ejpam-5887	667	60	4	4	NUM
ejpam-5887	667	61	)	)	PUNCT
ejpam-5887	667	62	2	2	NUM
ejpam-5887	667	63	;	;	PUNCT
ejpam-5887	667	64	j	j	PROPN
ejpam-5887	667	65	≡	≡	PROPN
ejpam-5887	667	66	2	2	NUM
ejpam-5887	667	67	,	,	PUNCT
ejpam-5887	667	68	3	3	NUM
ejpam-5887	667	69	(	(	PUNCT
ejpam-5887	667	70	mod	mod	NOUN
ejpam-5887	667	71	4	4	NUM
ejpam-5887	667	72	)	)	PUNCT
ejpam-5887	667	73	;	;	PUNCT
ejpam-5887	667	74	1	1	NUM
ejpam-5887	667	75	≤	≤	NUM
ejpam-5887	667	76	j	j	PROPN
ejpam-5887	667	77	≤	≤	NOUN
ejpam-5887	667	78	3t+	3t+	NUM
ejpam-5887	667	79	1	1	NUM
ejpam-5887	667	80	.	.	PUNCT
ejpam-5887	667	81	from	from	ADP
ejpam-5887	667	82	this	this	DET
ejpam-5887	667	83	labeling	labeling	NOUN
ejpam-5887	667	84	we	we	PRON
ejpam-5887	667	85	have	have	VERB
ejpam-5887	667	86	,	,	PUNCT
ejpam-5887	667	87	ef	ef	PROPN
ejpam-5887	667	88	(	(	PUNCT
ejpam-5887	667	89	i	i	NOUN
ejpam-5887	667	90	)	)	PUNCT
ejpam-5887	667	91	=	=	PRON
ejpam-5887	667	92	{	{	PUNCT
ejpam-5887	667	93	2t+	2t+	NUM
ejpam-5887	667	94	1	1	NUM
ejpam-5887	667	95	;	;	PUNCT
ejpam-5887	667	96	i	i	PRON
ejpam-5887	667	97	=	=	SYM
ejpam-5887	667	98	0	0	NUM
ejpam-5887	667	99	2t+	2t+	NUM
ejpam-5887	667	100	2	2	NUM
ejpam-5887	667	101	;	;	PUNCT
ejpam-5887	667	102	i	i	NOUN
ejpam-5887	667	103	=	=	NOUN
ejpam-5887	667	104	1	1	NUM
ejpam-5887	667	105	,	,	PUNCT
ejpam-5887	667	106	2	2	NUM
ejpam-5887	667	107	,	,	PUNCT
ejpam-5887	667	108	vf∗(i	vf∗(i	ADJ
ejpam-5887	667	109	)	)	PUNCT
ejpam-5887	667	110	=	=	SYM
ejpam-5887	667	111	{	{	PUNCT
ejpam-5887	667	112	2t+	2t+	NUM
ejpam-5887	667	113	1	1	NUM
ejpam-5887	667	114	;	;	PUNCT
ejpam-5887	667	115	i	i	PRON
ejpam-5887	667	116	=	=	NOUN
ejpam-5887	667	117	1	1	NUM
ejpam-5887	667	118	,	,	PUNCT
ejpam-5887	667	119	2	2	NUM
ejpam-5887	667	120	2t+	2t+	NUM
ejpam-5887	667	121	2	2	NUM
ejpam-5887	667	122	;	;	PUNCT
ejpam-5887	667	123	i	i	PROPN
ejpam-5887	667	124	=	=	NOUN
ejpam-5887	667	125	0	0	X
ejpam-5887	667	126	.	.	PUNCT
ejpam-5887	668	1	conversely	conversely	ADV
ejpam-5887	668	2	,	,	PUNCT
ejpam-5887	668	3	let	let	VERB
ejpam-5887	668	4	f	f	PRON
ejpam-5887	668	5	be	be	AUX
ejpam-5887	668	6	an	an	DET
ejpam-5887	668	7	edge	edge	NOUN
ejpam-5887	668	8	3	3	NUM
ejpam-5887	668	9	-	-	PUNCT
ejpam-5887	668	10	product	product	NOUN
ejpam-5887	668	11	cordial	cordial	ADJ
ejpam-5887	668	12	labeling	labeling	NOUN
ejpam-5887	668	13	of	of	ADP
ejpam-5887	668	14	p	p	NOUN
ejpam-5887	668	15	(	(	PUNCT
ejpam-5887	668	16	2.c3t+1	2.c3t+1	NUM
ejpam-5887	668	17	)	)	PUNCT
ejpam-5887	668	18	;	;	PUNCT
ejpam-5887	669	1	t	t	PROPN
ejpam-5887	669	2	≥	≥	NUM
ejpam-5887	669	3	1	1	NUM
ejpam-5887	669	4	.	.	PUNCT
ejpam-5887	670	1	then	then	ADV
ejpam-5887	670	2	,	,	PUNCT
ejpam-5887	670	3	ef	ef	PROPN
ejpam-5887	670	4	(	(	PUNCT
ejpam-5887	670	5	i	i	NOUN
ejpam-5887	670	6	)	)	PUNCT
ejpam-5887	670	7	=	=	PUNCT
ejpam-5887	670	8	2	2	NUM
ejpam-5887	670	9	t	t	NOUN
ejpam-5887	670	10	+	+	NOUN
ejpam-5887	670	11	1	1	NUM
ejpam-5887	670	12	(	(	PUNCT
ejpam-5887	670	13	i	i	NOUN
ejpam-5887	670	14	=	=	NOUN
ejpam-5887	670	15	0	0	NUM
ejpam-5887	670	16	,	,	PUNCT
ejpam-5887	670	17	1	1	NUM
ejpam-5887	670	18	,	,	PUNCT
ejpam-5887	670	19	2	2	NUM
ejpam-5887	670	20	)	)	PUNCT
ejpam-5887	670	21	and	and	CCONJ
ejpam-5887	670	22	vf∗(i	vf∗(i	ADJ
ejpam-5887	670	23	)	)	PUNCT
ejpam-5887	670	24	=	=	SYM
ejpam-5887	670	25	2	2	NUM
ejpam-5887	670	26	t	t	NOUN
ejpam-5887	670	27	or	or	CCONJ
ejpam-5887	670	28	2	2	NUM
ejpam-5887	670	29	t	t	NOUN
ejpam-5887	670	30	+	+	NOUN
ejpam-5887	670	31	1	1	NUM
ejpam-5887	670	32	(	(	PUNCT
ejpam-5887	670	33	i	i	NOUN
ejpam-5887	670	34	=	=	NOUN
ejpam-5887	670	35	0	0	NUM
ejpam-5887	670	36	,	,	PUNCT
ejpam-5887	670	37	1	1	NUM
ejpam-5887	670	38	,	,	PUNCT
ejpam-5887	670	39	2	2	NUM
ejpam-5887	670	40	)	)	PUNCT
ejpam-5887	670	41	.	.	PUNCT
ejpam-5887	671	1	clearly	clearly	ADV
ejpam-5887	671	2	,	,	PUNCT
ejpam-5887	671	3	ef	ef	PROPN
ejpam-5887	671	4	(	(	PUNCT
ejpam-5887	671	5	0	0	NUM
ejpam-5887	671	6	)	)	PUNCT
ejpam-5887	671	7	=	=	SYM
ejpam-5887	671	8	2	2	NUM
ejpam-5887	671	9	t	t	NOUN
ejpam-5887	671	10	+	+	NOUN
ejpam-5887	671	11	1	1	NUM
ejpam-5887	671	12	implies	imply	VERB
ejpam-5887	671	13	vf∗(0	vf∗(0	NOUN
ejpam-5887	671	14	)	)	PUNCT
ejpam-5887	671	15	≥	≥	NOUN
ejpam-5887	671	16	2	2	NUM
ejpam-5887	671	17	t	t	NOUN
ejpam-5887	671	18	+	+	CCONJ
ejpam-5887	671	19	2	2	NUM
ejpam-5887	671	20	>	>	SYM
ejpam-5887	671	21	2	2	NUM
ejpam-5887	671	22	t	t	NOUN
ejpam-5887	671	23	+	+	NOUN
ejpam-5887	671	24	1	1	NUM
ejpam-5887	671	25	,	,	PUNCT
ejpam-5887	671	26	which	which	PRON
ejpam-5887	671	27	is	be	AUX
ejpam-5887	671	28	a	a	DET
ejpam-5887	671	29	contradiction	contradiction	NOUN
ejpam-5887	671	30	.	.	PUNCT
ejpam-5887	672	1	hence	hence	ADV
ejpam-5887	672	2	,	,	PUNCT
ejpam-5887	672	3	p	p	X
ejpam-5887	672	4	(	(	PUNCT
ejpam-5887	672	5	2.c3t+1	2.c3t+1	NUM
ejpam-5887	672	6	)	)	PUNCT
ejpam-5887	672	7	;	;	PUNCT
ejpam-5887	672	8	t	t	PROPN
ejpam-5887	672	9	≥	≥	NUM
ejpam-5887	672	10	1	1	NUM
ejpam-5887	672	11	is	be	AUX
ejpam-5887	672	12	not	not	PART
ejpam-5887	672	13	an	an	DET
ejpam-5887	672	14	edge	edge	NOUN
ejpam-5887	672	15	3	3	NUM
ejpam-5887	672	16	-	-	PUNCT
ejpam-5887	672	17	product	product	NOUN
ejpam-5887	672	18	cordial	cordial	ADJ
ejpam-5887	672	19	graph	graph	NOUN
ejpam-5887	672	20	.	.	PUNCT
ejpam-5887	673	1	by	by	ADP
ejpam-5887	673	2	theorem	theorem	NOUN
ejpam-5887	673	3	21	21	NUM
ejpam-5887	673	4	,	,	PUNCT
ejpam-5887	673	5	p	p	X
ejpam-5887	673	6	(	(	PUNCT
ejpam-5887	673	7	2.c3	2.c3	PROPN
ejpam-5887	673	8	t	t	PROPN
ejpam-5887	673	9	)	)	PUNCT
ejpam-5887	673	10	;	;	PUNCT
ejpam-5887	673	11	t	t	PROPN
ejpam-5887	673	12	≥	≥	NUM
ejpam-5887	673	13	1	1	NUM
ejpam-5887	673	14	is	be	AUX
ejpam-5887	673	15	not	not	PART
ejpam-5887	673	16	an	an	DET
ejpam-5887	673	17	edge	edge	NOUN
ejpam-5887	673	18	3	3	NUM
ejpam-5887	673	19	-	-	PUNCT
ejpam-5887	673	20	product	product	NOUN
ejpam-5887	673	21	cordial	cordial	ADJ
ejpam-5887	673	22	graph	graph	NOUN
ejpam-5887	673	23	.	.	PUNCT
ejpam-5887	674	1	also	also	ADV
ejpam-5887	674	2	,	,	PUNCT
ejpam-5887	674	3	by	by	ADP
ejpam-5887	674	4	theorem	theorem	NOUN
ejpam-5887	674	5	20	20	NUM
ejpam-5887	674	6	,	,	PUNCT
ejpam-5887	674	7	p	p	X
ejpam-5887	674	8	(	(	PUNCT
ejpam-5887	674	9	n.cm	n.cm	PROPN
ejpam-5887	674	10	)	)	PUNCT
ejpam-5887	674	11	;	;	PUNCT
ejpam-5887	674	12	n	n	PRON
ejpam-5887	674	13	≥	≥	NOUN
ejpam-5887	674	14	3	3	NUM
ejpam-5887	674	15	is	be	AUX
ejpam-5887	674	16	not	not	PART
ejpam-5887	674	17	an	an	DET
ejpam-5887	674	18	edge	edge	NOUN
ejpam-5887	674	19	3	3	NUM
ejpam-5887	674	20	-	-	PUNCT
ejpam-5887	674	21	product	product	NOUN
ejpam-5887	674	22	cordial	cordial	ADJ
ejpam-5887	674	23	graph	graph	NOUN
ejpam-5887	674	24	.	.	PUNCT
ejpam-5887	674	25	example	example	NOUN
ejpam-5887	675	1	7	7	NUM
ejpam-5887	675	2	.	.	PUNCT
ejpam-5887	675	3	an	an	DET
ejpam-5887	675	4	edge	edge	NOUN
ejpam-5887	675	5	3	3	NUM
ejpam-5887	675	6	-	-	PUNCT
ejpam-5887	675	7	product	product	NOUN
ejpam-5887	675	8	cordial	cordial	ADJ
ejpam-5887	675	9	labeling	labeling	NOUN
ejpam-5887	675	10	of	of	ADP
ejpam-5887	675	11	p	p	NOUN
ejpam-5887	675	12	(	(	PUNCT
ejpam-5887	675	13	n.c8	n.c8	NOUN
ejpam-5887	675	14	)	)	PUNCT
ejpam-5887	675	15	is	be	AUX
ejpam-5887	675	16	shown	show	VERB
ejpam-5887	675	17	in	in	ADP
ejpam-5887	675	18	figure	figure	NOUN
ejpam-5887	675	19	7	7	NUM
ejpam-5887	675	20	.	.	NOUN
ejpam-5887	675	21	0	0	NUM
ejpam-5887	675	22	0	0	NUM
ejpam-5887	675	23	0	0	NUM
ejpam-5887	675	24	0	0	NUM
ejpam-5887	675	25	0	0	NUM
ejpam-5887	675	26	00	00	NUM
ejpam-5887	675	27	0	0	NUM
ejpam-5887	675	28	0	0	NUM
ejpam-5887	675	29	0	0	NUM
ejpam-5887	675	30	0	0	NUM
ejpam-5887	675	31	1	1	NUM
ejpam-5887	675	32	1	1	NUM
ejpam-5887	675	33	1	1	NUM
ejpam-5887	675	34	1	1	NUM
ejpam-5887	675	35	1	1	NUM
ejpam-5887	675	36	1	1	NUM
ejpam-5887	675	37	1	1	NUM
ejpam-5887	675	38	1	1	NUM
ejpam-5887	675	39	1	1	NUM
ejpam-5887	675	40	1	1	NUM
ejpam-5887	675	41	1	1	NUM
ejpam-5887	675	42	2	2	NUM
ejpam-5887	675	43	2	2	NUM
ejpam-5887	675	44	2	2	NUM
ejpam-5887	675	45	2	2	NUM
ejpam-5887	675	46	2	2	NUM
ejpam-5887	675	47	2	2	NUM
ejpam-5887	675	48	2	2	NUM
ejpam-5887	675	49	2	2	NUM
ejpam-5887	675	50	2	2	NUM
ejpam-5887	675	51	2	2	NUM
ejpam-5887	675	52	2	2	NUM
ejpam-5887	675	53	figure	figure	NOUN
ejpam-5887	675	54	7	7	NUM
ejpam-5887	675	55	:	:	PUNCT
ejpam-5887	675	56	edge	edge	VERB
ejpam-5887	675	57	3	3	NUM
ejpam-5887	675	58	-	-	PUNCT
ejpam-5887	675	59	product	product	NOUN
ejpam-5887	675	60	cordial	cordial	ADJ
ejpam-5887	675	61	labeling	labeling	NOUN
ejpam-5887	675	62	of	of	ADP
ejpam-5887	675	63	p	p	NOUN
ejpam-5887	675	64	(	(	PUNCT
ejpam-5887	675	65	n.c8	n.c8	NOUN
ejpam-5887	675	66	)	)	PUNCT
ejpam-5887	675	67	theorem	theorem	VERB
ejpam-5887	675	68	23	23	NUM
ejpam-5887	675	69	.	.	PUNCT
ejpam-5887	676	1	the	the	DET
ejpam-5887	676	2	path	path	PROPN
ejpam-5887	676	3	union	union	PROPN
ejpam-5887	676	4	of	of	ADP
ejpam-5887	676	5	cycle	cycle	NOUN
ejpam-5887	676	6	graph	graph	NOUN
ejpam-5887	676	7	p	p	X
ejpam-5887	676	8	(	(	PUNCT
ejpam-5887	676	9	n.cm	n.cm	PROPN
ejpam-5887	676	10	)	)	PUNCT
ejpam-5887	676	11	admits	admit	VERB
ejpam-5887	676	12	an	an	DET
ejpam-5887	676	13	edge	edge	NOUN
ejpam-5887	676	14	4	4	NUM
ejpam-5887	676	15	-	-	PUNCT
ejpam-5887	676	16	product	product	NOUN
ejpam-5887	676	17	cordial	cordial	ADJ
ejpam-5887	676	18	labeling	labeling	NOUN
ejpam-5887	676	19	if	if	SCONJ
ejpam-5887	676	20	and	and	CCONJ
ejpam-5887	676	21	only	only	ADV
ejpam-5887	676	22	if	if	SCONJ
ejpam-5887	676	23	n	n	PROPN
ejpam-5887	676	24	=	=	SYM
ejpam-5887	676	25	2	2	NUM
ejpam-5887	676	26	and	and	CCONJ
ejpam-5887	676	27	m	m	PROPN
ejpam-5887	676	28	=	=	NOUN
ejpam-5887	676	29	3	3	NUM
ejpam-5887	676	30	,	,	PUNCT
ejpam-5887	676	31	5	5	NUM
ejpam-5887	676	32	.	.	PUNCT
ejpam-5887	677	1	proof	proof	NOUN
ejpam-5887	677	2	.	.	PUNCT
ejpam-5887	678	1	let	let	VERB
ejpam-5887	678	2	the	the	DET
ejpam-5887	678	3	vertex	vertex	NOUN
ejpam-5887	678	4	and	and	CCONJ
ejpam-5887	678	5	edge	edge	NOUN
ejpam-5887	678	6	set	set	NOUN
ejpam-5887	678	7	of	of	ADP
ejpam-5887	678	8	p	p	PROPN
ejpam-5887	678	9	(	(	PUNCT
ejpam-5887	678	10	n.cm	n.cm	PROPN
ejpam-5887	678	11	)	)	PUNCT
ejpam-5887	678	12	be	be	VERB
ejpam-5887	678	13	v	v	ADP
ejpam-5887	678	14	(	(	PUNCT
ejpam-5887	678	15	p	p	X
ejpam-5887	678	16	(	(	PUNCT
ejpam-5887	678	17	n.cm	n.cm	ADJ
ejpam-5887	678	18	)	)	PUNCT
ejpam-5887	678	19	)	)	PUNCT
ejpam-5887	679	1	=	=	PRON
ejpam-5887	679	2	{	{	PUNCT
ejpam-5887	679	3	vji	vji	NOUN
ejpam-5887	679	4	;	;	PUNCT
ejpam-5887	679	5	1	1	NUM
ejpam-5887	679	6	≤	≤	NUM
ejpam-5887	679	7	i	i	PRON
ejpam-5887	679	8	≤	≤	PROPN
ejpam-5887	679	9	n	n	CCONJ
ejpam-5887	679	10	,	,	PUNCT
ejpam-5887	679	11	1	1	NUM
ejpam-5887	679	12	≤	≤	NUM
ejpam-5887	679	13	j	j	PROPN
ejpam-5887	679	14	≤	≤	PROPN
ejpam-5887	679	15	m	m	PROPN
ejpam-5887	679	16	}	}	PUNCT
ejpam-5887	679	17	and	and	CCONJ
ejpam-5887	679	18	e(p	e(p	PROPN
ejpam-5887	679	19	(	(	PUNCT
ejpam-5887	679	20	n.cm	n.cm	ADJ
ejpam-5887	679	21	)	)	PUNCT
ejpam-5887	679	22	)	)	PUNCT
ejpam-5887	680	1	=	=	PRON
ejpam-5887	680	2	{	{	PUNCT
ejpam-5887	680	3	v1i	v1i	NUM
ejpam-5887	680	4	v1i+1	v1i+1	NOUN
ejpam-5887	680	5	,	,	PUNCT
ejpam-5887	680	6	v	v	NOUN
ejpam-5887	680	7	j	j	NOUN
ejpam-5887	681	1	i	i	PRON
ejpam-5887	681	2	v	v	VERB
ejpam-5887	681	3	j+1	j+1	ADV
ejpam-5887	682	1	i	i	PRON
ejpam-5887	682	2	,	,	PUNCT
ejpam-5887	682	3	vjnv	vjnv	NOUN
ejpam-5887	682	4	j+1	j+1	ADJ
ejpam-5887	682	5	n	n	NOUN
ejpam-5887	682	6	,	,	PUNCT
ejpam-5887	682	7	vmi	vmi	PROPN
ejpam-5887	682	8	v1i	v1i	PROPN
ejpam-5887	682	9	,	,	PUNCT
ejpam-5887	682	10	v	v	NOUN
ejpam-5887	682	11	m	m	PROPN
ejpam-5887	682	12	n	n	PRON
ejpam-5887	682	13	v1n	v1n	ADJ
ejpam-5887	682	14	;	;	PUNCT
ejpam-5887	682	15	1	1	NUM
ejpam-5887	682	16	≤	≤	NUM
ejpam-5887	682	17	i	i	PRON
ejpam-5887	682	18	≤	≤	ADJ
ejpam-5887	682	19	n	n	CCONJ
ejpam-5887	682	20	−	−	PROPN
ejpam-5887	682	21	1	1	NUM
ejpam-5887	682	22	,	,	PUNCT
ejpam-5887	682	23	1	1	NUM
ejpam-5887	682	24	≤	≤	NUM
ejpam-5887	682	25	j	j	PROPN
ejpam-5887	682	26	≤	≤	PROPN
ejpam-5887	682	27	m−	m−	PROPN
ejpam-5887	682	28	1	1	NUM
ejpam-5887	682	29	}	}	PUNCT
ejpam-5887	682	30	respectively	respectively	ADV
ejpam-5887	682	31	.	.	PUNCT
ejpam-5887	683	1	define	define	VERB
ejpam-5887	683	2	f	f	PROPN
ejpam-5887	683	3	:	:	PUNCT
ejpam-5887	683	4	e(p	e(p	PROPN
ejpam-5887	683	5	(	(	PUNCT
ejpam-5887	683	6	2.cm	2.cm	NUM
ejpam-5887	683	7	)	)	PUNCT
ejpam-5887	683	8	→	→	SYM
ejpam-5887	683	9	{	{	PUNCT
ejpam-5887	683	10	0	0	NUM
ejpam-5887	683	11	,	,	PUNCT
ejpam-5887	683	12	1	1	NUM
ejpam-5887	683	13	,	,	PUNCT
ejpam-5887	683	14	2	2	NUM
ejpam-5887	683	15	,	,	PUNCT
ejpam-5887	683	16	3	3	NUM
ejpam-5887	683	17	}	}	PUNCT
ejpam-5887	683	18	for	for	ADP
ejpam-5887	683	19	m	m	PROPN
ejpam-5887	683	20	=	=	SYM
ejpam-5887	683	21	3	3	NUM
ejpam-5887	683	22	,	,	PUNCT
ejpam-5887	683	23	5	5	NUM
ejpam-5887	683	24	as	as	SCONJ
ejpam-5887	683	25	follows	follow	VERB
ejpam-5887	683	26	:	:	PUNCT
ejpam-5887	683	27	f(v11v	f(v11v	ADJ
ejpam-5887	683	28	1	1	NUM
ejpam-5887	683	29	2	2	NUM
ejpam-5887	683	30	)	)	PUNCT
ejpam-5887	683	31	=	=	SYM
ejpam-5887	683	32	2	2	NUM
ejpam-5887	683	33	,	,	PUNCT
ejpam-5887	683	34	f(vji	f(vji	NOUN
ejpam-5887	683	35	v	v	NOUN
ejpam-5887	683	36	j+1	j+1	PROPN
ejpam-5887	683	37	i	i	NOUN
ejpam-5887	683	38	)	)	PUNCT
ejpam-5887	683	39	=	=	PUNCT
ejpam-5887	684	1			X
ejpam-5887	684	2	0	0	NUM
ejpam-5887	684	3	;	;	PUNCT
ejpam-5887	684	4	i	i	PRON
ejpam-5887	684	5	=	=	NOUN
ejpam-5887	684	6	1	1	NUM
ejpam-5887	684	7	,	,	PUNCT
ejpam-5887	684	8	j	j	NOUN
ejpam-5887	684	9	=	=	SYM
ejpam-5887	684	10	1	1	NUM
ejpam-5887	684	11	1	1	NUM
ejpam-5887	684	12	;	;	PUNCT
ejpam-5887	684	13	i	i	PRON
ejpam-5887	684	14	=	=	NOUN
ejpam-5887	684	15	1	1	NUM
ejpam-5887	684	16	,	,	PUNCT
ejpam-5887	684	17	j	j	NOUN
ejpam-5887	684	18	=	=	SYM
ejpam-5887	684	19	2	2	NUM
ejpam-5887	684	20	3	3	NUM
ejpam-5887	684	21	;	;	PUNCT
ejpam-5887	684	22	i	i	PRON
ejpam-5887	684	23	=	=	SYM
ejpam-5887	684	24	2	2	NUM
ejpam-5887	684	25	,	,	PUNCT
ejpam-5887	684	26	1	1	NUM
ejpam-5887	684	27	≤	≤	NUM
ejpam-5887	684	28	j	j	PROPN
ejpam-5887	684	29	≤	≤	ADV
ejpam-5887	684	30	2	2	NUM
ejpam-5887	684	31	;	;	PUNCT
ejpam-5887	684	32	m	m	VERB
ejpam-5887	684	33	=	=	SYM
ejpam-5887	684	34	3	3	NUM
ejpam-5887	684	35	,	,	PUNCT
ejpam-5887	684	36	f(vji	f(vji	NOUN
ejpam-5887	684	37	v	v	NOUN
ejpam-5887	684	38	j+1	j+1	PROPN
ejpam-5887	684	39	i	i	NOUN
ejpam-5887	684	40	)	)	PUNCT
ejpam-5887	684	41	=	=	PUNCT
ejpam-5887	685	1			PROPN
ejpam-5887	685	2	0	0	NUM
ejpam-5887	685	3	;	;	PUNCT
ejpam-5887	686	1	i	i	PRON
ejpam-5887	686	2	=	=	NOUN
ejpam-5887	686	3	1	1	NUM
ejpam-5887	686	4	,	,	PUNCT
ejpam-5887	686	5	1	1	NUM
ejpam-5887	686	6	≤	≤	NUM
ejpam-5887	686	7	j	j	PROPN
ejpam-5887	686	8	≤	≤	ADV
ejpam-5887	686	9	2	2	NUM
ejpam-5887	686	10	1	1	NUM
ejpam-5887	686	11	;	;	PUNCT
ejpam-5887	686	12	i	i	PRON
ejpam-5887	686	13	=	=	NOUN
ejpam-5887	686	14	1	1	NUM
ejpam-5887	686	15	,	,	PUNCT
ejpam-5887	686	16	j	j	NOUN
ejpam-5887	686	17	=	=	NOUN
ejpam-5887	686	18	4	4	NUM
ejpam-5887	686	19	;	;	PUNCT
ejpam-5887	686	20	i	i	PRON
ejpam-5887	686	21	=	=	SYM
ejpam-5887	686	22	2	2	NUM
ejpam-5887	686	23	,	,	PUNCT
ejpam-5887	686	24	2	2	NUM
ejpam-5887	686	25	≤	≤	NUM
ejpam-5887	686	26	j	j	PROPN
ejpam-5887	686	27	≤	≤	ADV
ejpam-5887	686	28	3	3	NUM
ejpam-5887	686	29	2	2	NUM
ejpam-5887	686	30	;	;	PUNCT
ejpam-5887	686	31	i	i	NOUN
ejpam-5887	686	32	=	=	NOUN
ejpam-5887	686	33	1	1	NUM
ejpam-5887	686	34	,	,	PUNCT
ejpam-5887	686	35	j	j	PROPN
ejpam-5887	686	36	=	=	SYM
ejpam-5887	686	37	3	3	NUM
ejpam-5887	686	38	3	3	NUM
ejpam-5887	686	39	;	;	PUNCT
ejpam-5887	686	40	i	i	PRON
ejpam-5887	686	41	=	=	NOUN
ejpam-5887	686	42	2	2	NUM
ejpam-5887	686	43	,	,	PUNCT
ejpam-5887	686	44	j	j	PROPN
ejpam-5887	686	45	=	=	SYM
ejpam-5887	686	46	1	1	NUM
ejpam-5887	686	47	,	,	PUNCT
ejpam-5887	686	48	4	4	NUM
ejpam-5887	686	49	;	;	PUNCT
ejpam-5887	686	50	m	m	VERB
ejpam-5887	686	51	=	=	SYM
ejpam-5887	686	52	5	5	NUM
ejpam-5887	686	53	,	,	PUNCT
ejpam-5887	686	54	n.	n.	NOUN
ejpam-5887	686	55	m.	m.	NOUN
ejpam-5887	686	56	noureldeen	noureldeen	NOUN
ejpam-5887	686	57	et	et	PROPN
ejpam-5887	686	58	al	al	PROPN
ejpam-5887	686	59	.	.	PUNCT
ejpam-5887	686	60	/	/	SYM
ejpam-5887	686	61	eur	eur	PROPN
ejpam-5887	686	62	.	.	PUNCT
ejpam-5887	687	1	j.	j.	PROPN
ejpam-5887	687	2	pure	pure	PROPN
ejpam-5887	687	3	appl	appl	PROPN
ejpam-5887	687	4	.	.	PROPN
ejpam-5887	687	5	math	math	PROPN
ejpam-5887	687	6	,	,	PUNCT
ejpam-5887	687	7	18	18	NUM
ejpam-5887	687	8	(	(	PUNCT
ejpam-5887	687	9	2	2	NUM
ejpam-5887	687	10	)	)	PUNCT
ejpam-5887	687	11	(	(	PUNCT
ejpam-5887	687	12	2025	2025	NUM
ejpam-5887	687	13	)	)	PUNCT
ejpam-5887	687	14	,	,	PUNCT
ejpam-5887	687	15	5887	5887	NUM
ejpam-5887	687	16	19	19	NUM
ejpam-5887	687	17	of	of	ADP
ejpam-5887	687	18	21	21	NUM
ejpam-5887	687	19	f(vmi	f(vmi	ADJ
ejpam-5887	687	20	v1i	v1i	NOUN
ejpam-5887	687	21	)	)	PUNCT
ejpam-5887	687	22	=	=	PUNCT
ejpam-5887	688	1			PUNCT
ejpam-5887	688	2	1	1	NUM
ejpam-5887	688	3	;	;	PUNCT
ejpam-5887	688	4	i	i	NOUN
ejpam-5887	688	5	=	=	NOUN
ejpam-5887	688	6	2	2	NUM
ejpam-5887	688	7	,	,	PUNCT
ejpam-5887	688	8	m	m	VERB
ejpam-5887	688	9	=	=	NOUN
ejpam-5887	688	10	3	3	NUM
ejpam-5887	688	11	2	2	NUM
ejpam-5887	688	12	;	;	PUNCT
ejpam-5887	688	13	i	i	NOUN
ejpam-5887	688	14	=	=	NOUN
ejpam-5887	688	15	1	1	NUM
ejpam-5887	688	16	,	,	PUNCT
ejpam-5887	688	17	m	m	VERB
ejpam-5887	688	18	=	=	NOUN
ejpam-5887	688	19	3	3	NUM
ejpam-5887	688	20	,	,	PUNCT
ejpam-5887	688	21	5	5	NUM
ejpam-5887	688	22	3	3	NUM
ejpam-5887	688	23	;	;	PUNCT
ejpam-5887	688	24	i	i	PRON
ejpam-5887	688	25	=	=	NOUN
ejpam-5887	688	26	2	2	NUM
ejpam-5887	688	27	,	,	PUNCT
ejpam-5887	688	28	m	m	VERB
ejpam-5887	688	29	=	=	NOUN
ejpam-5887	688	30	5	5	NUM
ejpam-5887	688	31	.	.	PUNCT
ejpam-5887	688	32	from	from	ADP
ejpam-5887	688	33	this	this	DET
ejpam-5887	688	34	labeling	labeling	NOUN
ejpam-5887	688	35	we	we	PRON
ejpam-5887	688	36	have	have	VERB
ejpam-5887	688	37	,	,	PUNCT
ejpam-5887	688	38	ef	ef	PROPN
ejpam-5887	688	39	(	(	PUNCT
ejpam-5887	688	40	i	i	NOUN
ejpam-5887	688	41	)	)	PUNCT
ejpam-5887	688	42	=	=	PRON
ejpam-5887	688	43	{	{	PUNCT
ejpam-5887	688	44	2⌊m4	2⌊m4	NUM
ejpam-5887	688	45	⌋	⌋	NOUN
ejpam-5887	688	46	;	;	PUNCT
ejpam-5887	689	1	i	i	PRON
ejpam-5887	689	2	=	=	NOUN
ejpam-5887	689	3	0	0	NUM
ejpam-5887	689	4	2⌊m4	2⌊m4	NUM
ejpam-5887	689	5	⌋+	⌋+	ADJ
ejpam-5887	689	6	1	1	NUM
ejpam-5887	689	7	;	;	PUNCT
ejpam-5887	689	8	1	1	NUM
ejpam-5887	689	9	≤	≤	NUM
ejpam-5887	689	10	i	i	X
ejpam-5887	689	11	≤	≤	NOUN
ejpam-5887	689	12	3	3	NUM
ejpam-5887	689	13	,	,	PUNCT
ejpam-5887	689	14	vf∗(i	vf∗(i	ADJ
ejpam-5887	689	15	)	)	PUNCT
ejpam-5887	689	16	=	=	VERB
ejpam-5887	689	17	{	{	PUNCT
ejpam-5887	689	18	2⌊m4	2⌊m4	NUM
ejpam-5887	689	19	⌋	⌋	NOUN
ejpam-5887	689	20	;	;	PUNCT
ejpam-5887	689	21	i	i	NOUN
ejpam-5887	689	22	=	=	NOUN
ejpam-5887	689	23	1	1	NUM
ejpam-5887	689	24	,	,	PUNCT
ejpam-5887	689	25	3	3	NUM
ejpam-5887	689	26	2⌊m4	2⌊m4	NUM
ejpam-5887	689	27	⌋+	⌋+	ADJ
ejpam-5887	689	28	1	1	NUM
ejpam-5887	689	29	;	;	PUNCT
ejpam-5887	689	30	i	i	PRON
ejpam-5887	689	31	=	=	NOUN
ejpam-5887	689	32	0	0	NUM
ejpam-5887	689	33	,	,	PUNCT
ejpam-5887	689	34	2	2	NUM
ejpam-5887	689	35	.	.	PUNCT
ejpam-5887	690	1	hence	hence	ADV
ejpam-5887	690	2	,	,	PUNCT
ejpam-5887	690	3	p	p	X
ejpam-5887	690	4	(	(	PUNCT
ejpam-5887	690	5	2.c3	2.c3	NUM
ejpam-5887	690	6	)	)	PUNCT
ejpam-5887	690	7	and	and	CCONJ
ejpam-5887	690	8	p	p	X
ejpam-5887	690	9	(	(	PUNCT
ejpam-5887	690	10	2.c5	2.c5	ADV
ejpam-5887	690	11	)	)	PUNCT
ejpam-5887	690	12	are	be	AUX
ejpam-5887	690	13	edge	edge	VERB
ejpam-5887	690	14	4	4	NUM
ejpam-5887	690	15	-	-	PUNCT
ejpam-5887	690	16	product	product	NOUN
ejpam-5887	690	17	cordial	cordial	ADJ
ejpam-5887	690	18	graphs	graph	NOUN
ejpam-5887	690	19	.	.	PUNCT
ejpam-5887	691	1	to	to	PART
ejpam-5887	691	2	prove	prove	VERB
ejpam-5887	691	3	the	the	DET
ejpam-5887	691	4	converse	converse	NOUN
ejpam-5887	691	5	part	part	NOUN
ejpam-5887	691	6	,	,	PUNCT
ejpam-5887	691	7	we	we	PRON
ejpam-5887	691	8	consider	consider	VERB
ejpam-5887	691	9	the	the	DET
ejpam-5887	691	10	following	follow	VERB
ejpam-5887	691	11	two	two	NUM
ejpam-5887	691	12	cases	case	NOUN
ejpam-5887	691	13	.	.	PUNCT
ejpam-5887	692	1	case(i	case(i	NOUN
ejpam-5887	692	2	):	):	PUNCT
ejpam-5887	692	3	if	if	SCONJ
ejpam-5887	692	4	n	n	PROPN
ejpam-5887	692	5	=	=	SYM
ejpam-5887	692	6	2	2	NUM
ejpam-5887	692	7	and	and	CCONJ
ejpam-5887	692	8	m	m	NOUN
ejpam-5887	692	9	=	=	VERB
ejpam-5887	692	10	4	4	NUM
ejpam-5887	692	11	t	t	NOUN
ejpam-5887	692	12	+	+	CCONJ
ejpam-5887	692	13	r	r	NOUN
ejpam-5887	692	14	,	,	PUNCT
ejpam-5887	692	15	where	where	SCONJ
ejpam-5887	692	16	1	1	NUM
ejpam-5887	692	17	≤	≤	NOUN
ejpam-5887	692	18	r	r	NOUN
ejpam-5887	692	19	≤	≤	NUM
ejpam-5887	692	20	3	3	NUM
ejpam-5887	692	21	,	,	PUNCT
ejpam-5887	692	22	t	t	PROPN
ejpam-5887	692	23	≥	≥	NUM
ejpam-5887	692	24	2	2	NUM
ejpam-5887	692	25	for	for	ADP
ejpam-5887	692	26	r	r	NOUN
ejpam-5887	692	27	=	=	SYM
ejpam-5887	692	28	1	1	NUM
ejpam-5887	692	29	and	and	CCONJ
ejpam-5887	692	30	t	t	PROPN
ejpam-5887	692	31	≥	≥	NUM
ejpam-5887	692	32	1	1	NUM
ejpam-5887	692	33	for	for	ADP
ejpam-5887	692	34	2	2	NUM
ejpam-5887	692	35	≤	≤	NOUN
ejpam-5887	692	36	r	r	NOUN
ejpam-5887	692	37	≤	≤	NUM
ejpam-5887	692	38	3	3	NUM
ejpam-5887	692	39	,	,	PUNCT
ejpam-5887	692	40	then	then	ADV
ejpam-5887	692	41	|v	|v	PROPN
ejpam-5887	692	42	|	|	ADV
ejpam-5887	692	43	=	=	SYM
ejpam-5887	692	44	8t+	8t+	NUM
ejpam-5887	692	45	2r	2r	NUM
ejpam-5887	692	46	and	and	CCONJ
ejpam-5887	692	47	|e|	|e|	NOUN
ejpam-5887	692	48	=	=	PUNCT
ejpam-5887	692	49	8t+	8t+	NUM
ejpam-5887	692	50	2r	2r	NUM
ejpam-5887	692	51	+	+	CCONJ
ejpam-5887	692	52	1	1	X
ejpam-5887	692	53	.	.	PUNCT
ejpam-5887	692	54	let	let	VERB
ejpam-5887	692	55	f	f	PRON
ejpam-5887	692	56	be	be	AUX
ejpam-5887	692	57	an	an	DET
ejpam-5887	692	58	edge	edge	NOUN
ejpam-5887	692	59	4	4	NUM
ejpam-5887	692	60	-	-	PUNCT
ejpam-5887	692	61	product	product	NOUN
ejpam-5887	692	62	cordial	cordial	ADJ
ejpam-5887	692	63	labeling	labeling	NOUN
ejpam-5887	692	64	of	of	ADP
ejpam-5887	692	65	the	the	DET
ejpam-5887	692	66	graph	graph	NOUN
ejpam-5887	692	67	p	p	X
ejpam-5887	692	68	(	(	PUNCT
ejpam-5887	692	69	2.cm	2.cm	NUM
ejpam-5887	692	70	)	)	PUNCT
ejpam-5887	692	71	.	.	PUNCT
ejpam-5887	693	1	then	then	ADV
ejpam-5887	693	2	we	we	PRON
ejpam-5887	693	3	have	have	VERB
ejpam-5887	693	4	,	,	PUNCT
ejpam-5887	693	5	ef	ef	PROPN
ejpam-5887	693	6	(	(	PUNCT
ejpam-5887	693	7	i	i	NOUN
ejpam-5887	693	8	)	)	PUNCT
ejpam-5887	693	9	=	=	PRON
ejpam-5887	693	10	{	{	PUNCT
ejpam-5887	693	11	2	2	NUM
ejpam-5887	693	12	t	t	NOUN
ejpam-5887	693	13	or	or	CCONJ
ejpam-5887	693	14	2t+	2t+	NUM
ejpam-5887	693	15	1	1	NUM
ejpam-5887	693	16	;	;	PUNCT
ejpam-5887	693	17	r	r	NOUN
ejpam-5887	693	18	=	=	SYM
ejpam-5887	693	19	1	1	NUM
ejpam-5887	693	20	2t+	2t+	NUM
ejpam-5887	693	21	1	1	NUM
ejpam-5887	693	22	or	or	CCONJ
ejpam-5887	693	23	2t+	2t+	NUM
ejpam-5887	693	24	2	2	NUM
ejpam-5887	693	25	;	;	PUNCT
ejpam-5887	693	26	r	r	NOUN
ejpam-5887	693	27	=	=	SYM
ejpam-5887	693	28	2	2	NUM
ejpam-5887	693	29	,	,	PUNCT
ejpam-5887	693	30	3	3	NUM
ejpam-5887	693	31	,	,	PUNCT
ejpam-5887	693	32	vf∗(i	vf∗(i	ADJ
ejpam-5887	693	33	)	)	PUNCT
ejpam-5887	693	34	=	=	PUNCT
ejpam-5887	693	35			PUNCT
ejpam-5887	693	36	2	2	NUM
ejpam-5887	693	37	t	t	NOUN
ejpam-5887	693	38	or	or	CCONJ
ejpam-5887	693	39	2t+	2t+	NUM
ejpam-5887	693	40	1	1	NUM
ejpam-5887	693	41	;	;	PUNCT
ejpam-5887	693	42	r	r	NOUN
ejpam-5887	693	43	=	=	SYM
ejpam-5887	693	44	1	1	NUM
ejpam-5887	693	45	2t+	2t+	NUM
ejpam-5887	693	46	1	1	NUM
ejpam-5887	693	47	;	;	PUNCT
ejpam-5887	694	1	r	r	NOUN
ejpam-5887	694	2	=	=	SYM
ejpam-5887	694	3	2	2	NUM
ejpam-5887	694	4	2t+	2t+	NUM
ejpam-5887	694	5	1	1	NUM
ejpam-5887	694	6	or	or	CCONJ
ejpam-5887	694	7	2t+	2t+	NUM
ejpam-5887	694	8	2	2	NUM
ejpam-5887	694	9	;	;	PUNCT
ejpam-5887	694	10	r	r	NOUN
ejpam-5887	694	11	=	=	SYM
ejpam-5887	694	12	3	3	X
ejpam-5887	694	13	.	.	X
ejpam-5887	694	14	for	for	ADP
ejpam-5887	694	15	r	r	NOUN
ejpam-5887	694	16	=	=	SYM
ejpam-5887	694	17	1	1	NUM
ejpam-5887	694	18	,	,	PUNCT
ejpam-5887	694	19	we	we	PRON
ejpam-5887	694	20	must	must	AUX
ejpam-5887	694	21	have	have	VERB
ejpam-5887	694	22	the	the	DET
ejpam-5887	694	23	following	follow	VERB
ejpam-5887	694	24	conditions	condition	NOUN
ejpam-5887	694	25	.	.	PUNCT
ejpam-5887	695	1	(	(	PUNCT
ejpam-5887	695	2	i	i	NOUN
ejpam-5887	695	3	)	)	PUNCT
ejpam-5887	695	4	ef	ef	PROPN
ejpam-5887	695	5	(	(	PUNCT
ejpam-5887	695	6	0	0	NUM
ejpam-5887	695	7	)	)	PUNCT
ejpam-5887	695	8	=	=	SYM
ejpam-5887	695	9	2	2	NUM
ejpam-5887	695	10	t	t	NOUN
ejpam-5887	695	11	,	,	PUNCT
ejpam-5887	695	12	(	(	PUNCT
ejpam-5887	695	13	ii	ii	NOUN
ejpam-5887	695	14	)	)	PUNCT
ejpam-5887	695	15	two	two	NUM
ejpam-5887	695	16	adjacent	adjacent	ADJ
ejpam-5887	695	17	edges	edge	NOUN
ejpam-5887	695	18	can	can	AUX
ejpam-5887	695	19	not	not	PART
ejpam-5887	695	20	be	be	AUX
ejpam-5887	695	21	labeled	label	VERB
ejpam-5887	695	22	with	with	ADP
ejpam-5887	695	23	2	2	NUM
ejpam-5887	695	24	,	,	PUNCT
ejpam-5887	695	25	(	(	PUNCT
ejpam-5887	695	26	iii	iii	NOUN
ejpam-5887	695	27	)	)	PUNCT
ejpam-5887	695	28	0	0	NUM
ejpam-5887	695	29	must	must	AUX
ejpam-5887	695	30	be	be	AUX
ejpam-5887	695	31	assigned	assign	VERB
ejpam-5887	695	32	consecutively	consecutively	ADV
ejpam-5887	695	33	otherwise	otherwise	ADV
ejpam-5887	695	34	vf∗(0	vf∗(0	NOUN
ejpam-5887	695	35	)	)	PUNCT
ejpam-5887	695	36	≥	≥	NOUN
ejpam-5887	695	37	2t+2	2t+2	PROPN
ejpam-5887	695	38	.	.	PUNCT
ejpam-5887	696	1	hence	hence	ADV
ejpam-5887	696	2	,	,	PUNCT
ejpam-5887	696	3	ef	ef	PROPN
ejpam-5887	696	4	(	(	PUNCT
ejpam-5887	696	5	i	i	NOUN
ejpam-5887	696	6	)	)	PUNCT
ejpam-5887	696	7	=	=	SYM
ejpam-5887	697	1	2t+1	2t+1	PROPN
ejpam-5887	697	2	(	(	PUNCT
ejpam-5887	697	3	i	i	NOUN
ejpam-5887	697	4	=	=	NOUN
ejpam-5887	697	5	1	1	NUM
ejpam-5887	697	6	,	,	PUNCT
ejpam-5887	697	7	2	2	NUM
ejpam-5887	697	8	,	,	PUNCT
ejpam-5887	697	9	3	3	NUM
ejpam-5887	697	10	)	)	PUNCT
ejpam-5887	697	11	and	and	CCONJ
ejpam-5887	697	12	2t+1	2t+1	PROPN
ejpam-5887	697	13	non	non	NOUN
ejpam-5887	697	14	adjacent	adjacent	ADJ
ejpam-5887	697	15	edges	edge	NOUN
ejpam-5887	697	16	must	must	AUX
ejpam-5887	697	17	be	be	AUX
ejpam-5887	697	18	labeled	label	VERB
ejpam-5887	697	19	with	with	ADP
ejpam-5887	697	20	2	2	NUM
ejpam-5887	697	21	.	.	PUNCT
ejpam-5887	698	1	this	this	PRON
ejpam-5887	698	2	implies	imply	VERB
ejpam-5887	698	3	vf∗(2	vf∗(2	ADJ
ejpam-5887	698	4	)	)	PUNCT
ejpam-5887	698	5	≥	≥	NOUN
ejpam-5887	698	6	2t+	2t+	NUM
ejpam-5887	698	7	2	2	NUM
ejpam-5887	698	8	.	.	PUNCT
ejpam-5887	698	9	by	by	ADP
ejpam-5887	698	10	similar	similar	ADJ
ejpam-5887	698	11	argument	argument	NOUN
ejpam-5887	698	12	,	,	PUNCT
ejpam-5887	698	13	for	for	ADP
ejpam-5887	698	14	r	r	NOUN
ejpam-5887	698	15	=	=	SYM
ejpam-5887	698	16	3	3	NUM
ejpam-5887	698	17	,	,	PUNCT
ejpam-5887	698	18	we	we	PRON
ejpam-5887	698	19	get	get	VERB
ejpam-5887	698	20	ef	ef	X
ejpam-5887	698	21	(	(	PUNCT
ejpam-5887	698	22	0	0	NUM
ejpam-5887	698	23	)	)	PUNCT
ejpam-5887	698	24	=	=	SYM
ejpam-5887	698	25	2t+1	2t+1	PROPN
ejpam-5887	698	26	,	,	PUNCT
ejpam-5887	698	27	ef	ef	PROPN
ejpam-5887	698	28	(	(	PUNCT
ejpam-5887	698	29	i	i	NOUN
ejpam-5887	698	30	)	)	PUNCT
ejpam-5887	699	1	=	=	SYM
ejpam-5887	699	2	2t+2	2t+2	PROPN
ejpam-5887	699	3	(	(	PUNCT
ejpam-5887	699	4	i	i	NOUN
ejpam-5887	699	5	=	=	NOUN
ejpam-5887	699	6	1	1	NUM
ejpam-5887	699	7	,	,	PUNCT
ejpam-5887	699	8	2	2	NUM
ejpam-5887	699	9	,	,	PUNCT
ejpam-5887	699	10	3	3	NUM
ejpam-5887	699	11	)	)	PUNCT
ejpam-5887	699	12	and	and	CCONJ
ejpam-5887	699	13	vf∗(2	vf∗(2	ADJ
ejpam-5887	699	14	)	)	PUNCT
ejpam-5887	699	15	≥	≥	NOUN
ejpam-5887	699	16	2t+3	2t+3	NUM
ejpam-5887	699	17	.	.	PUNCT
ejpam-5887	700	1	for	for	ADP
ejpam-5887	700	2	r	r	NOUN
ejpam-5887	700	3	=	=	SYM
ejpam-5887	700	4	2	2	NUM
ejpam-5887	700	5	,	,	PUNCT
ejpam-5887	700	6	ef	ef	X
ejpam-5887	700	7	(	(	PUNCT
ejpam-5887	700	8	0	0	NUM
ejpam-5887	700	9	)	)	PUNCT
ejpam-5887	700	10	=	=	SYM
ejpam-5887	700	11	2t+1	2t+1	PROPN
ejpam-5887	700	12	implies	imply	VERB
ejpam-5887	700	13	vf∗(0	vf∗(0	NOUN
ejpam-5887	700	14	)	)	PUNCT
ejpam-5887	700	15	≥	≥	NOUN
ejpam-5887	700	16	2t+	2t+	NUM
ejpam-5887	700	17	2	2	NUM
ejpam-5887	700	18	.	.	PUNCT
ejpam-5887	700	19	therefore	therefore	ADV
ejpam-5887	700	20	in	in	ADP
ejpam-5887	700	21	all	all	DET
ejpam-5887	700	22	the	the	DET
ejpam-5887	700	23	cases	case	NOUN
ejpam-5887	700	24	,	,	PUNCT
ejpam-5887	700	25	we	we	PRON
ejpam-5887	700	26	obtain	obtain	VERB
ejpam-5887	700	27	|vf∗(0)−	|vf∗(0)−	PROPN
ejpam-5887	700	28	vf∗(2)|	vf∗(2)|	NOUN
ejpam-5887	700	29	≥	≥	NUM
ejpam-5887	700	30	2	2	NUM
ejpam-5887	700	31	,	,	PUNCT
ejpam-5887	700	32	which	which	PRON
ejpam-5887	700	33	is	be	AUX
ejpam-5887	700	34	a	a	DET
ejpam-5887	700	35	contradiction	contradiction	NOUN
ejpam-5887	700	36	.	.	PUNCT
ejpam-5887	701	1	case(ii	case(ii	ADJ
ejpam-5887	701	2	):	):	PUNCT
ejpam-5887	701	3	if	if	SCONJ
ejpam-5887	701	4	n	n	PROPN
ejpam-5887	701	5	=	=	SYM
ejpam-5887	701	6	3	3	NUM
ejpam-5887	701	7	and	and	CCONJ
ejpam-5887	701	8	m	m	NOUN
ejpam-5887	701	9	=	=	NOUN
ejpam-5887	701	10	4	4	NUM
ejpam-5887	701	11	t	t	NOUN
ejpam-5887	701	12	+	+	CCONJ
ejpam-5887	701	13	r	r	NOUN
ejpam-5887	701	14	,	,	PUNCT
ejpam-5887	701	15	where	where	SCONJ
ejpam-5887	701	16	1	1	NUM
ejpam-5887	701	17	≤	≤	NOUN
ejpam-5887	701	18	r	r	NOUN
ejpam-5887	701	19	≤	≤	NUM
ejpam-5887	701	20	3	3	NUM
ejpam-5887	701	21	,	,	PUNCT
ejpam-5887	701	22	t	t	PROPN
ejpam-5887	701	23	≥	≥	NUM
ejpam-5887	701	24	1	1	NUM
ejpam-5887	701	25	for	for	ADP
ejpam-5887	701	26	1	1	NUM
ejpam-5887	701	27	≤	≤	NOUN
ejpam-5887	701	28	r	r	NOUN
ejpam-5887	701	29	≤	≤	NUM
ejpam-5887	701	30	2	2	NUM
ejpam-5887	701	31	and	and	CCONJ
ejpam-5887	701	32	t	t	PROPN
ejpam-5887	701	33	≥	≥	NOUN
ejpam-5887	701	34	0	0	NUM
ejpam-5887	701	35	for	for	ADP
ejpam-5887	701	36	r	r	NOUN
ejpam-5887	701	37	=	=	SYM
ejpam-5887	701	38	3	3	NUM
ejpam-5887	701	39	,	,	PUNCT
ejpam-5887	701	40	then	then	ADV
ejpam-5887	701	41	|v	|v	PROPN
ejpam-5887	701	42	|	|	ADV
ejpam-5887	701	43	=	=	SYM
ejpam-5887	701	44	12t+	12t+	NUM
ejpam-5887	701	45	3r	3r	NUM
ejpam-5887	701	46	and	and	CCONJ
ejpam-5887	701	47	|e|	|e|	NOUN
ejpam-5887	701	48	=	=	PUNCT
ejpam-5887	701	49	12t+	12t+	NUM
ejpam-5887	701	50	3r	3r	NUM
ejpam-5887	701	51	+	+	CCONJ
ejpam-5887	701	52	2	2	X
ejpam-5887	701	53	.	.	X
ejpam-5887	701	54	let	let	VERB
ejpam-5887	701	55	g	g	PRON
ejpam-5887	701	56	be	be	AUX
ejpam-5887	701	57	an	an	DET
ejpam-5887	701	58	edge	edge	NOUN
ejpam-5887	701	59	4	4	NUM
ejpam-5887	701	60	-	-	PUNCT
ejpam-5887	701	61	product	product	NOUN
ejpam-5887	701	62	cordial	cordial	ADJ
ejpam-5887	701	63	labeling	labeling	NOUN
ejpam-5887	701	64	of	of	ADP
ejpam-5887	701	65	the	the	DET
ejpam-5887	701	66	graph	graph	NOUN
ejpam-5887	701	67	p	p	X
ejpam-5887	701	68	(	(	PUNCT
ejpam-5887	701	69	3.cm	3.cm	NUM
ejpam-5887	701	70	)	)	PUNCT
ejpam-5887	701	71	.	.	PUNCT
ejpam-5887	702	1	then	then	ADV
ejpam-5887	702	2	we	we	PRON
ejpam-5887	702	3	have	have	VERB
ejpam-5887	702	4	,	,	PUNCT
ejpam-5887	702	5	eg(i	eg(i	X
ejpam-5887	702	6	)	)	PUNCT
ejpam-5887	702	7	=	=	PUNCT
ejpam-5887	703	1			PUNCT
ejpam-5887	703	2	3t+	3t+	NUM
ejpam-5887	703	3	1	1	NUM
ejpam-5887	703	4	or	or	CCONJ
ejpam-5887	703	5	3t+	3t+	NUM
ejpam-5887	703	6	2	2	NUM
ejpam-5887	703	7	;	;	PUNCT
ejpam-5887	703	8	r	r	NOUN
ejpam-5887	703	9	=	=	SYM
ejpam-5887	703	10	1	1	NUM
ejpam-5887	703	11	3t+	3t+	NUM
ejpam-5887	703	12	2	2	NUM
ejpam-5887	703	13	;	;	PUNCT
ejpam-5887	703	14	r	r	NOUN
ejpam-5887	703	15	=	=	SYM
ejpam-5887	703	16	2	2	NUM
ejpam-5887	703	17	3t+	3t+	NUM
ejpam-5887	703	18	2	2	NUM
ejpam-5887	703	19	or	or	CCONJ
ejpam-5887	703	20	3t+	3t+	NUM
ejpam-5887	703	21	3	3	NUM
ejpam-5887	703	22	;	;	PUNCT
ejpam-5887	703	23	r	r	NOUN
ejpam-5887	703	24	=	=	SYM
ejpam-5887	703	25	3	3	NUM
ejpam-5887	703	26	,	,	PUNCT
ejpam-5887	703	27	vg∗(i	vg∗(i	ADJ
ejpam-5887	703	28	)	)	PUNCT
ejpam-5887	703	29	=	=	PUNCT
ejpam-5887	703	30			PUNCT
ejpam-5887	703	31	3	3	NUM
ejpam-5887	703	32	t	t	NOUN
ejpam-5887	703	33	or	or	CCONJ
ejpam-5887	703	34	3t+	3t+	NUM
ejpam-5887	703	35	1	1	NUM
ejpam-5887	703	36	;	;	PUNCT
ejpam-5887	703	37	r	r	NOUN
ejpam-5887	703	38	=	=	SYM
ejpam-5887	703	39	1	1	NUM
ejpam-5887	703	40	3t+	3t+	NUM
ejpam-5887	703	41	1	1	NUM
ejpam-5887	703	42	or	or	CCONJ
ejpam-5887	703	43	3t+	3t+	NUM
ejpam-5887	703	44	2	2	NUM
ejpam-5887	703	45	;	;	PUNCT
ejpam-5887	703	46	r	r	NOUN
ejpam-5887	703	47	=	=	SYM
ejpam-5887	703	48	2	2	NUM
ejpam-5887	703	49	3t+	3t+	NUM
ejpam-5887	703	50	2	2	NUM
ejpam-5887	703	51	or	or	CCONJ
ejpam-5887	703	52	3t+	3t+	NUM
ejpam-5887	703	53	3	3	NUM
ejpam-5887	703	54	;	;	PUNCT
ejpam-5887	703	55	r	r	NOUN
ejpam-5887	703	56	=	=	SYM
ejpam-5887	703	57	3	3	X
ejpam-5887	703	58	.	.	NOUN
ejpam-5887	704	1	for	for	ADP
ejpam-5887	704	2	1	1	NUM
ejpam-5887	704	3	≤	≤	NOUN
ejpam-5887	704	4	r	r	NOUN
ejpam-5887	704	5	≤	≤	NUM
ejpam-5887	704	6	2	2	NUM
ejpam-5887	704	7	,	,	PUNCT
ejpam-5887	704	8	eg(0	eg(0	NOUN
ejpam-5887	704	9	)	)	PUNCT
ejpam-5887	704	10	=	=	SYM
ejpam-5887	704	11	3t+	3t+	NUM
ejpam-5887	704	12	r	r	NOUN
ejpam-5887	704	13	implies	imply	VERB
ejpam-5887	704	14	vg∗(0	vg∗(0	NOUN
ejpam-5887	704	15	)	)	PUNCT
ejpam-5887	704	16	≥	≥	NOUN
ejpam-5887	704	17	3t+	3t+	NUM
ejpam-5887	704	18	r	r	NOUN
ejpam-5887	704	19	+	+	NOUN
ejpam-5887	704	20	1	1	NUM
ejpam-5887	704	21	.	.	X
ejpam-5887	704	22	for	for	ADP
ejpam-5887	704	23	r	r	NOUN
ejpam-5887	704	24	=	=	SYM
ejpam-5887	704	25	3	3	NUM
ejpam-5887	704	26	,	,	PUNCT
ejpam-5887	704	27	as	as	ADP
ejpam-5887	704	28	in	in	ADP
ejpam-5887	704	29	case(i	case(i	NOUN
ejpam-5887	704	30	)	)	PUNCT
ejpam-5887	704	31	we	we	PRON
ejpam-5887	704	32	get	get	VERB
ejpam-5887	704	33	,	,	PUNCT
ejpam-5887	704	34	eg(0	eg(0	NOUN
ejpam-5887	704	35	)	)	PUNCT
ejpam-5887	704	36	=	=	PUNCT
ejpam-5887	704	37	3	3	NUM
ejpam-5887	704	38	t	t	NOUN
ejpam-5887	704	39	+	+	NOUN
ejpam-5887	704	40	2	2	NUM
ejpam-5887	704	41	,	,	PUNCT
ejpam-5887	704	42	eg(i	eg(i	X
ejpam-5887	704	43	)	)	PUNCT
ejpam-5887	704	44	=	=	PUNCT
ejpam-5887	705	1	3	3	NUM
ejpam-5887	705	2	t	t	NOUN
ejpam-5887	705	3	+	+	NOUN
ejpam-5887	705	4	3	3	NUM
ejpam-5887	705	5	(	(	PUNCT
ejpam-5887	705	6	i	i	NOUN
ejpam-5887	705	7	=	=	NOUN
ejpam-5887	705	8	1	1	NUM
ejpam-5887	705	9	,	,	PUNCT
ejpam-5887	705	10	2	2	NUM
ejpam-5887	705	11	,	,	PUNCT
ejpam-5887	705	12	3	3	NUM
ejpam-5887	705	13	)	)	PUNCT
ejpam-5887	705	14	and	and	CCONJ
ejpam-5887	705	15	vg∗(i	vg∗(i	ADJ
ejpam-5887	705	16	)	)	PUNCT
ejpam-5887	705	17	=	=	SYM
ejpam-5887	706	1	3	3	NUM
ejpam-5887	706	2	t	t	NOUN
ejpam-5887	706	3	+	+	NOUN
ejpam-5887	706	4	2	2	NUM
ejpam-5887	706	5	(	(	PUNCT
ejpam-5887	706	6	i	i	NOUN
ejpam-5887	706	7	=	=	NOUN
ejpam-5887	706	8	1	1	NUM
ejpam-5887	706	9	,	,	PUNCT
ejpam-5887	706	10	2	2	NUM
ejpam-5887	706	11	,	,	PUNCT
ejpam-5887	706	12	3	3	NUM
ejpam-5887	706	13	)	)	PUNCT
ejpam-5887	706	14	,	,	PUNCT
ejpam-5887	706	15	which	which	PRON
ejpam-5887	706	16	result	result	VERB
ejpam-5887	706	17	vg∗(2	vg∗(2	PROPN
ejpam-5887	706	18	)	)	PUNCT
ejpam-5887	706	19	≥	≥	NOUN
ejpam-5887	706	20	3	3	NUM
ejpam-5887	706	21	t	t	NOUN
ejpam-5887	706	22	+	+	NOUN
ejpam-5887	706	23	3	3	X
ejpam-5887	706	24	.	.	X
ejpam-5887	706	25	therefore	therefore	ADV
ejpam-5887	706	26	,	,	PUNCT
ejpam-5887	706	27	in	in	ADP
ejpam-5887	706	28	all	all	DET
ejpam-5887	706	29	the	the	DET
ejpam-5887	706	30	cases	case	NOUN
ejpam-5887	706	31	,	,	PUNCT
ejpam-5887	706	32	we	we	PRON
ejpam-5887	706	33	obtain	obtain	VERB
ejpam-5887	706	34	|vg∗(0	|vg∗(0	NOUN
ejpam-5887	706	35	)	)	PUNCT
ejpam-5887	706	36	−	−	PROPN
ejpam-5887	706	37	vg∗(2)|	vg∗(2)|	NOUN
ejpam-5887	706	38	≥	≥	NOUN
ejpam-5887	706	39	2	2	NUM
ejpam-5887	706	40	,	,	PUNCT
ejpam-5887	706	41	which	which	PRON
ejpam-5887	706	42	is	be	AUX
ejpam-5887	706	43	a	a	DET
ejpam-5887	706	44	contradiction	contradiction	NOUN
ejpam-5887	706	45	.	.	PUNCT
ejpam-5887	707	1	by	by	ADP
ejpam-5887	707	2	theorem	theorem	NOUN
ejpam-5887	707	3	21	21	NUM
ejpam-5887	707	4	,	,	PUNCT
ejpam-5887	707	5	p	p	X
ejpam-5887	707	6	(	(	PUNCT
ejpam-5887	707	7	n.c4	n.c4	NUM
ejpam-5887	707	8	t	t	PROPN
ejpam-5887	707	9	)	)	PUNCT
ejpam-5887	707	10	;	;	PUNCT
ejpam-5887	707	11	t	t	PROPN
ejpam-5887	707	12	≥	≥	NUM
ejpam-5887	707	13	1	1	NUM
ejpam-5887	707	14	is	be	AUX
ejpam-5887	707	15	not	not	PART
ejpam-5887	707	16	an	an	DET
ejpam-5887	707	17	edge	edge	NOUN
ejpam-5887	707	18	4	4	NUM
ejpam-5887	707	19	-	-	PUNCT
ejpam-5887	707	20	product	product	NOUN
ejpam-5887	707	21	cordial	cordial	ADJ
ejpam-5887	707	22	graph	graph	NOUN
ejpam-5887	707	23	.	.	PUNCT
ejpam-5887	708	1	also	also	ADV
ejpam-5887	708	2	,	,	PUNCT
ejpam-5887	708	3	by	by	ADP
ejpam-5887	708	4	theorem	theorem	NOUN
ejpam-5887	708	5	20	20	NUM
ejpam-5887	708	6	,	,	PUNCT
ejpam-5887	708	7	p	p	X
ejpam-5887	708	8	(	(	PUNCT
ejpam-5887	708	9	n.cm	n.cm	PROPN
ejpam-5887	708	10	)	)	PUNCT
ejpam-5887	708	11	;	;	PUNCT
ejpam-5887	709	1	n	n	PRON
ejpam-5887	709	2	≥	≥	NOUN
ejpam-5887	709	3	4	4	NUM
ejpam-5887	709	4	is	be	AUX
ejpam-5887	709	5	not	not	PART
ejpam-5887	709	6	an	an	DET
ejpam-5887	709	7	edge	edge	NOUN
ejpam-5887	709	8	4	4	NUM
ejpam-5887	709	9	-	-	PUNCT
ejpam-5887	709	10	product	product	NOUN
ejpam-5887	709	11	cordial	cordial	ADJ
ejpam-5887	709	12	graph	graph	NOUN
ejpam-5887	709	13	.	.	PUNCT
ejpam-5887	710	1	n.	n.	PROPN
ejpam-5887	710	2	m.	m.	NOUN
ejpam-5887	710	3	noureldeen	noureldeen	INTJ
ejpam-5887	710	4	et	et	PROPN
ejpam-5887	710	5	al	al	PROPN
ejpam-5887	710	6	.	.	PUNCT
ejpam-5887	710	7	/	/	SYM
ejpam-5887	710	8	eur	eur	PROPN
ejpam-5887	710	9	.	.	PUNCT
ejpam-5887	711	1	j.	j.	PROPN
ejpam-5887	711	2	pure	pure	PROPN
ejpam-5887	711	3	appl	appl	PROPN
ejpam-5887	711	4	.	.	PROPN
ejpam-5887	711	5	math	math	PROPN
ejpam-5887	711	6	,	,	PUNCT
ejpam-5887	711	7	18	18	NUM
ejpam-5887	711	8	(	(	PUNCT
ejpam-5887	711	9	2	2	NUM
ejpam-5887	711	10	)	)	PUNCT
ejpam-5887	711	11	(	(	PUNCT
ejpam-5887	711	12	2025	2025	NUM
ejpam-5887	711	13	)	)	PUNCT
ejpam-5887	711	14	,	,	PUNCT
ejpam-5887	711	15	5887	5887	NUM
ejpam-5887	711	16	20	20	NUM
ejpam-5887	711	17	of	of	ADP
ejpam-5887	711	18	21	21	NUM
ejpam-5887	711	19	references	reference	NOUN
ejpam-5887	711	20	[	[	X
ejpam-5887	711	21	1	1	NUM
ejpam-5887	711	22	]	]	PUNCT
ejpam-5887	711	23	r.	r.	PROPN
ejpam-5887	711	24	manoharan	manoharan	PROPN
ejpam-5887	711	25	a.	a.	PROPN
ejpam-5887	711	26	anto	anto	PROPN
ejpam-5887	711	27	cathrin	cathrin	PROPN
ejpam-5887	711	28	aanisha	aanisha	PROPN
ejpam-5887	711	29	.	.	PUNCT
ejpam-5887	712	1	mean	mean	VERB
ejpam-5887	712	2	cordial	cordial	ADJ
ejpam-5887	712	3	labeling	labeling	NOUN
ejpam-5887	712	4	in	in	ADP
ejpam-5887	712	5	graph	graph	NOUN
ejpam-5887	712	6	representations	representation	NOUN
ejpam-5887	712	7	of	of	ADP
ejpam-5887	712	8	human	human	ADJ
ejpam-5887	712	9	anatomy	anatomy	NOUN
ejpam-5887	712	10	and	and	CCONJ
ejpam-5887	712	11	circular	circular	ADJ
ejpam-5887	712	12	systems	system	NOUN
ejpam-5887	712	13	.	.	PUNCT
ejpam-5887	713	1	communications	communication	NOUN
ejpam-5887	713	2	on	on	ADP
ejpam-5887	713	3	applied	apply	VERB
ejpam-5887	713	4	nonlinear	nonlinear	ADJ
ejpam-5887	713	5	analysis	analysis	NOUN
ejpam-5887	713	6	,	,	PUNCT
ejpam-5887	713	7	31(7):453–458	31(7):453–458	NOUN
ejpam-5887	713	8	,	,	PUNCT
ejpam-5887	713	9	2024	2024	NUM
ejpam-5887	713	10	.	.	PUNCT
ejpam-5887	714	1	[	[	X
ejpam-5887	714	2	2	2	NUM
ejpam-5887	714	3	]	]	PUNCT
ejpam-5887	714	4	c.	c.	PROPN
ejpam-5887	714	5	m.	m.	PROPN
ejpam-5887	714	6	barasara	barasara	PROPN
ejpam-5887	714	7	.	.	PUNCT
ejpam-5887	715	1	edge	edge	NOUN
ejpam-5887	715	2	and	and	CCONJ
ejpam-5887	715	3	total	total	ADJ
ejpam-5887	715	4	edge	edge	NOUN
ejpam-5887	715	5	product	product	NOUN
ejpam-5887	715	6	cordial	cordial	ADJ
ejpam-5887	715	7	labeling	labeling	NOUN
ejpam-5887	715	8	of	of	ADP
ejpam-5887	715	9	some	some	DET
ejpam-5887	715	10	new	new	ADJ
ejpam-5887	715	11	graphs	graph	NOUN
ejpam-5887	715	12	.	.	PUNCT
ejpam-5887	716	1	international	international	ADJ
ejpam-5887	716	2	journal	journal	PROPN
ejpam-5887	716	3	of	of	ADP
ejpam-5887	716	4	engineering	engineering	NOUN
ejpam-5887	716	5	,	,	PUNCT
ejpam-5887	716	6	science	science	NOUN
ejpam-5887	716	7	and	and	CCONJ
ejpam-5887	716	8	mathematics	mathematic	NOUN
ejpam-5887	716	9	,	,	PUNCT
ejpam-5887	716	10	7(2):263–273	7(2):263–273	X
ejpam-5887	716	11	,	,	PUNCT
ejpam-5887	716	12	2018	2018	NUM
ejpam-5887	716	13	.	.	PUNCT
ejpam-5887	717	1	[	[	X
ejpam-5887	717	2	3	3	NUM
ejpam-5887	717	3	]	]	X
ejpam-5887	717	4	i.	i.	PROPN
ejpam-5887	717	5	cahit	cahit	PROPN
ejpam-5887	717	6	.	.	PUNCT
ejpam-5887	718	1	cordial	cordial	ADJ
ejpam-5887	718	2	graphs	graph	NOUN
ejpam-5887	718	3	:	:	PUNCT
ejpam-5887	718	4	a	a	DET
ejpam-5887	718	5	weaker	weak	ADJ
ejpam-5887	718	6	version	version	NOUN
ejpam-5887	718	7	of	of	ADP
ejpam-5887	718	8	graceful	graceful	ADJ
ejpam-5887	718	9	and	and	CCONJ
ejpam-5887	718	10	harmonious	harmonious	ADJ
ejpam-5887	718	11	graphs	graph	NOUN
ejpam-5887	718	12	.	.	PUNCT
ejpam-5887	719	1	ars	ar	NOUN
ejpam-5887	719	2	combin	combin	PROPN
ejpam-5887	719	3	.	.	PROPN
ejpam-5887	719	4	,	,	PUNCT
ejpam-5887	719	5	23:201–207	23:201–207	PROPN
ejpam-5887	719	6	,	,	PUNCT
ejpam-5887	719	7	1987	1987	NUM
ejpam-5887	719	8	.	.	PUNCT
ejpam-5887	720	1	[	[	X
ejpam-5887	720	2	4	4	X
ejpam-5887	720	3	]	]	PUNCT
ejpam-5887	720	4	j.	j.	PROPN
ejpam-5887	720	5	a.	a.	PROPN
ejpam-5887	720	6	gallian	gallian	PROPN
ejpam-5887	720	7	.	.	PUNCT
ejpam-5887	721	1	a	a	DET
ejpam-5887	721	2	dynamic	dynamic	ADJ
ejpam-5887	721	3	survey	survey	NOUN
ejpam-5887	721	4	of	of	ADP
ejpam-5887	721	5	graph	graph	NOUN
ejpam-5887	721	6	labeling	labeling	NOUN
ejpam-5887	721	7	.	.	PUNCT
ejpam-5887	722	1	the	the	DET
ejpam-5887	722	2	electronic	electronic	ADJ
ejpam-5887	722	3	journal	journal	NOUN
ejpam-5887	722	4	of	of	ADP
ejpam-5887	722	5	combinatorics	combinatoric	NOUN
ejpam-5887	722	6	,	,	PUNCT
ejpam-5887	722	7	#	#	NOUN
ejpam-5887	722	8	ds6	ds6	NOUN
ejpam-5887	722	9	,	,	PUNCT
ejpam-5887	722	10	2023	2023	NUM
ejpam-5887	722	11	.	.	PUNCT
ejpam-5887	723	1	[	[	X
ejpam-5887	723	2	5	5	X
ejpam-5887	723	3	]	]	PUNCT
ejpam-5887	723	4	e.	e.	PROPN
ejpam-5887	723	5	roshdy	roshdy	PROPN
ejpam-5887	723	6	m.	m.	PROPN
ejpam-5887	723	7	aboshady	aboshady	PROPN
ejpam-5887	723	8	,	,	PUNCT
ejpam-5887	723	9	r.	r.	PROPN
ejpam-5887	723	10	elbarkouky	elbarkouky	PROPN
ejpam-5887	723	11	and	and	CCONJ
ejpam-5887	723	12	m.	m.	PROPN
ejpam-5887	723	13	abdel	abdel	PROPN
ejpam-5887	723	14	-	-	PUNCT
ejpam-5887	723	15	azim	azim	PROPN
ejpam-5887	723	16	seoud	seoud	NOUN
ejpam-5887	723	17	.	.	PUNCT
ejpam-5887	724	1	further	further	ADJ
ejpam-5887	724	2	results	result	NOUN
ejpam-5887	724	3	on	on	ADP
ejpam-5887	724	4	edge	edge	NOUN
ejpam-5887	724	5	product	product	NOUN
ejpam-5887	724	6	cordial	cordial	ADJ
ejpam-5887	724	7	labeling	labeling	NOUN
ejpam-5887	724	8	.	.	PUNCT
ejpam-5887	725	1	proceedings	proceeding	NOUN
ejpam-5887	725	2	of	of	ADP
ejpam-5887	725	3	the	the	DET
ejpam-5887	725	4	pakistan	pakistan	PROPN
ejpam-5887	725	5	academy	academy	PROPN
ejpam-5887	725	6	of	of	ADP
ejpam-5887	725	7	sciences	sciences	PROPN
ejpam-5887	725	8	,	,	PUNCT
ejpam-5887	725	9	57(4):23–32	57(4):23–32	NUM
ejpam-5887	725	10	,	,	PUNCT
ejpam-5887	725	11	2020	2020	NUM
ejpam-5887	725	12	.	.	PUNCT
ejpam-5887	726	1	[	[	X
ejpam-5887	726	2	6	6	NUM
ejpam-5887	726	3	]	]	PUNCT
ejpam-5887	726	4	s.	s.	PROPN
ejpam-5887	726	5	somasundaram	somasundaram	PROPN
ejpam-5887	726	6	m.	m.	PROPN
ejpam-5887	726	7	sundaram	sundaram	PROPN
ejpam-5887	726	8	,	,	PUNCT
ejpam-5887	726	9	r.	r.	PROPN
ejpam-5887	726	10	ponraj	ponraj	PROPN
ejpam-5887	726	11	.	.	PUNCT
ejpam-5887	727	1	product	product	NOUN
ejpam-5887	727	2	cordial	cordial	ADJ
ejpam-5887	727	3	labeling	labeling	NOUN
ejpam-5887	727	4	of	of	ADP
ejpam-5887	727	5	graphs	graph	NOUN
ejpam-5887	727	6	.	.	PUNCT
ejpam-5887	728	1	bulletin	bulletin	NOUN
ejpam-5887	728	2	of	of	ADP
ejpam-5887	728	3	pure	pure	ADJ
ejpam-5887	728	4	and	and	CCONJ
ejpam-5887	728	5	applied	applied	ADJ
ejpam-5887	728	6	sciences	science	NOUN
ejpam-5887	728	7	,	,	PUNCT
ejpam-5887	728	8	23e(1):155–163	23e(1):155–163	NUM
ejpam-5887	728	9	,	,	PUNCT
ejpam-5887	728	10	2004	2004	NUM
ejpam-5887	728	11	.	.	PUNCT
ejpam-5887	729	1	[	[	X
ejpam-5887	729	2	7	7	X
ejpam-5887	729	3	]	]	PUNCT
ejpam-5887	729	4	andrea	andrea	PROPN
ejpam-5887	729	5	semaničová-feňovč́ıková	semaničová-feňovč́ıková	PROPN
ejpam-5887	729	6	martin	martin	PROPN
ejpam-5887	729	7	bača	bača	PROPN
ejpam-5887	729	8	,	,	PUNCT
ejpam-5887	729	9	muhammad	muhammad	PROPN
ejpam-5887	729	10	irfan	irfan	PROPN
ejpam-5887	729	11	.	.	PUNCT
ejpam-5887	730	1	on	on	ADP
ejpam-5887	730	2	3	3	NUM
ejpam-5887	730	3	-	-	PUNCT
ejpam-5887	730	4	total	total	ADJ
ejpam-5887	730	5	edge	edge	NOUN
ejpam-5887	730	6	product	product	NOUN
ejpam-5887	730	7	cordial	cordial	ADJ
ejpam-5887	730	8	labeling	labeling	NOUN
ejpam-5887	730	9	of	of	ADP
ejpam-5887	730	10	a	a	DET
ejpam-5887	730	11	carbon	carbon	NOUN
ejpam-5887	730	12	nanotube	nanotube	NOUN
ejpam-5887	730	13	network	network	NOUN
ejpam-5887	730	14	.	.	PUNCT
ejpam-5887	731	1	akce	akce	PROPN
ejpam-5887	731	2	international	international	PROPN
ejpam-5887	731	3	journal	journal	NOUN
ejpam-5887	731	4	of	of	ADP
ejpam-5887	731	5	graphs	graph	NOUN
ejpam-5887	731	6	and	and	CCONJ
ejpam-5887	731	7	combinatorics	combinatoric	NOUN
ejpam-5887	731	8	,	,	PUNCT
ejpam-5887	731	9	16:310–318	16:310–318	NUM
ejpam-5887	731	10	,	,	PUNCT
ejpam-5887	731	11	2019	2019	NUM
ejpam-5887	731	12	.	.	PUNCT
ejpam-5887	732	1	[	[	X
ejpam-5887	732	2	8	8	NUM
ejpam-5887	732	3	]	]	PUNCT
ejpam-5887	732	4	andrea	andrea	PROPN
ejpam-5887	732	5	semaničová-feňovč́ıková	semaničová-feňovč́ıková	PROPN
ejpam-5887	732	6	p.	p.	PROPN
ejpam-5887	732	7	jeyanthi	jeyanthi	PROPN
ejpam-5887	732	8	,	,	PUNCT
ejpam-5887	732	9	k.	k.	PROPN
ejpam-5887	732	10	jeya	jeya	PROPN
ejpam-5887	732	11	daisy	daisy	PROPN
ejpam-5887	732	12	.	.	PUNCT
ejpam-5887	733	1	zk	zk	PROPN
ejpam-5887	733	2	-	-	PUNCT
ejpam-5887	733	3	magic	magic	ADJ
ejpam-5887	733	4	labeling	labeling	NOUN
ejpam-5887	733	5	of	of	ADP
ejpam-5887	733	6	path	path	NOUN
ejpam-5887	733	7	union	union	PROPN
ejpam-5887	733	8	of	of	ADP
ejpam-5887	733	9	graphs	graph	NOUN
ejpam-5887	733	10	.	.	PUNCT
ejpam-5887	734	1	cubo	cubo	VERB
ejpam-5887	734	2	a	a	DET
ejpam-5887	734	3	mathematical	mathematical	ADJ
ejpam-5887	734	4	journal	journal	NOUN
ejpam-5887	734	5	,	,	PUNCT
ejpam-5887	734	6	21:15–35	21:15–35	NUM
ejpam-5887	734	7	,	,	PUNCT
ejpam-5887	734	8	2019	2019	NUM
ejpam-5887	734	9	.	.	PUNCT
ejpam-5887	735	1	[	[	X
ejpam-5887	735	2	9	9	NUM
ejpam-5887	735	3	]	]	PUNCT
ejpam-5887	735	4	r.	r.	X
ejpam-5887	735	5	rajeswari	rajeswari	PROPN
ejpam-5887	735	6	r.	r.	PROPN
ejpam-5887	735	7	thamizharasi	thamizharasi	PROPN
ejpam-5887	735	8	.	.	PUNCT
ejpam-5887	735	9	edge	edge	PROPN
ejpam-5887	735	10	product	product	NOUN
ejpam-5887	735	11	cordial	cordial	ADJ
ejpam-5887	735	12	labeling	labeling	NOUN
ejpam-5887	735	13	and	and	CCONJ
ejpam-5887	735	14	total	total	ADJ
ejpam-5887	735	15	magic	magic	ADJ
ejpam-5887	735	16	cordial	cordial	ADJ
ejpam-5887	735	17	labeling	labeling	NOUN
ejpam-5887	735	18	of	of	ADP
ejpam-5887	735	19	regular	regular	ADJ
ejpam-5887	735	20	digraphs	digraph	NOUN
ejpam-5887	735	21	.	.	PUNCT
ejpam-5887	736	1	international	international	ADJ
ejpam-5887	736	2	journal	journal	NOUN
ejpam-5887	736	3	on	on	ADP
ejpam-5887	736	4	information	information	NOUN
ejpam-5887	736	5	sciences	science	NOUN
ejpam-5887	736	6	and	and	CCONJ
ejpam-5887	736	7	computing	computing	NOUN
ejpam-5887	736	8	,	,	PUNCT
ejpam-5887	736	9	9(2):1–4	9(2):1–4	NUM
ejpam-5887	736	10	,	,	PUNCT
ejpam-5887	736	11	2015	2015	NUM
ejpam-5887	736	12	.	.	PUNCT
ejpam-5887	737	1	[	[	X
ejpam-5887	737	2	10	10	NUM
ejpam-5887	737	3	]	]	X
ejpam-5887	737	4	a.	a.	PROPN
ejpam-5887	737	5	rosa	rosa	PROPN
ejpam-5887	737	6	.	.	PUNCT
ejpam-5887	738	1	on	on	ADP
ejpam-5887	738	2	certain	certain	ADJ
ejpam-5887	738	3	valuations	valuation	NOUN
ejpam-5887	738	4	of	of	ADP
ejpam-5887	738	5	the	the	DET
ejpam-5887	738	6	vertices	vertex	NOUN
ejpam-5887	738	7	of	of	ADP
ejpam-5887	738	8	a	a	DET
ejpam-5887	738	9	graph	graph	NOUN
ejpam-5887	738	10	,	,	PUNCT
ejpam-5887	738	11	theory	theory	NOUN
ejpam-5887	738	12	of	of	ADP
ejpam-5887	738	13	graphs	graph	NOUN
ejpam-5887	738	14	.	.	PUNCT
ejpam-5887	739	1	internat	internat	PROPN
ejpam-5887	739	2	.	.	PUNCT
ejpam-5887	740	1	sympos	sympos	PROPN
ejpam-5887	740	2	.	.	PUNCT
ejpam-5887	740	3	,	,	PUNCT
ejpam-5887	740	4	pages	page	NOUN
ejpam-5887	740	5	349–355	349–355	NUM
ejpam-5887	740	6	,	,	PUNCT
ejpam-5887	740	7	1966	1966	NUM
ejpam-5887	740	8	.	.	PUNCT
ejpam-5887	741	1	[	[	X
ejpam-5887	741	2	11	11	NUM
ejpam-5887	741	3	]	]	X
ejpam-5887	741	4	c.	c.	PROPN
ejpam-5887	741	5	m.	m.	PROPN
ejpam-5887	741	6	barasara	barasara	PROPN
ejpam-5887	741	7	s.	s.	PROPN
ejpam-5887	741	8	k.	k.	PROPN
ejpam-5887	742	1	vaidya	vaidya	PROPN
ejpam-5887	742	2	.	.	PROPN
ejpam-5887	742	3	edge	edge	PROPN
ejpam-5887	742	4	product	product	NOUN
ejpam-5887	742	5	cordial	cordial	ADJ
ejpam-5887	742	6	labeling	labeling	NOUN
ejpam-5887	742	7	of	of	ADP
ejpam-5887	742	8	graphs	graph	NOUN
ejpam-5887	742	9	.	.	PUNCT
ejpam-5887	743	1	journal	journal	NOUN
ejpam-5887	743	2	of	of	ADP
ejpam-5887	743	3	mathematical	mathematical	ADJ
ejpam-5887	743	4	and	and	CCONJ
ejpam-5887	743	5	computational	computational	ADJ
ejpam-5887	743	6	science	science	NOUN
ejpam-5887	743	7	,	,	PUNCT
ejpam-5887	743	8	2(5):1436–1450	2(5):1436–1450	NUM
ejpam-5887	743	9	,	,	PUNCT
ejpam-5887	743	10	2012	2012	NUM
ejpam-5887	743	11	.	.	PUNCT
ejpam-5887	744	1	[	[	X
ejpam-5887	744	2	12	12	NUM
ejpam-5887	744	3	]	]	X
ejpam-5887	744	4	c.	c.	PROPN
ejpam-5887	744	5	m.	m.	PROPN
ejpam-5887	744	6	barasara	barasara	PROPN
ejpam-5887	744	7	s.	s.	PROPN
ejpam-5887	744	8	k.	k.	PROPN
ejpam-5887	745	1	vaidya	vaidya	PROPN
ejpam-5887	745	2	.	.	PROPN
ejpam-5887	745	3	edge	edge	PROPN
ejpam-5887	745	4	product	product	NOUN
ejpam-5887	745	5	cordial	cordial	ADJ
ejpam-5887	745	6	labeling	labeling	NOUN
ejpam-5887	745	7	in	in	ADP
ejpam-5887	745	8	the	the	DET
ejpam-5887	745	9	context	context	NOUN
ejpam-5887	745	10	of	of	ADP
ejpam-5887	745	11	some	some	DET
ejpam-5887	745	12	graph	graph	NOUN
ejpam-5887	745	13	operations	operation	NOUN
ejpam-5887	745	14	.	.	PUNCT
ejpam-5887	746	1	international	international	ADJ
ejpam-5887	746	2	journal	journal	PROPN
ejpam-5887	746	3	of	of	ADP
ejpam-5887	746	4	mathematics	mathematic	NOUN
ejpam-5887	746	5	and	and	CCONJ
ejpam-5887	746	6	scientific	scientific	ADJ
ejpam-5887	746	7	computing	computing	NOUN
ejpam-5887	746	8	,	,	PUNCT
ejpam-5887	746	9	3(1):4–7	3(1):4–7	NUM
ejpam-5887	746	10	,	,	PUNCT
ejpam-5887	746	11	2013	2013	NUM
ejpam-5887	746	12	.	.	PUNCT
ejpam-5887	747	1	[	[	X
ejpam-5887	747	2	13	13	NUM
ejpam-5887	747	3	]	]	X
ejpam-5887	747	4	c.	c.	PROPN
ejpam-5887	747	5	m.	m.	PROPN
ejpam-5887	747	6	barasara	barasara	PROPN
ejpam-5887	747	7	s.	s.	PROPN
ejpam-5887	747	8	k.	k.	PROPN
ejpam-5887	747	9	vaidya	vaidya	PROPN
ejpam-5887	747	10	.	.	PUNCT
ejpam-5887	748	1	some	some	DET
ejpam-5887	748	2	edge	edge	NOUN
ejpam-5887	748	3	product	product	NOUN
ejpam-5887	748	4	cordial	cordial	ADJ
ejpam-5887	748	5	graphs	graph	NOUN
ejpam-5887	748	6	.	.	PUNCT
ejpam-5887	749	1	international	international	ADJ
ejpam-5887	749	2	journal	journal	NOUN
ejpam-5887	749	3	of	of	ADP
ejpam-5887	749	4	mathematics	mathematic	NOUN
ejpam-5887	749	5	and	and	CCONJ
ejpam-5887	749	6	soft	soft	ADJ
ejpam-5887	749	7	computing	computing	NOUN
ejpam-5887	749	8	,	,	PUNCT
ejpam-5887	749	9	3(3):49–53	3(3):49–53	NUM
ejpam-5887	749	10	,	,	PUNCT
ejpam-5887	749	11	2013	2013	NUM
ejpam-5887	749	12	.	.	PUNCT
ejpam-5887	750	1	[	[	X
ejpam-5887	750	2	14	14	NUM
ejpam-5887	750	3	]	]	X
ejpam-5887	750	4	c.	c.	PROPN
ejpam-5887	750	5	m.	m.	PROPN
ejpam-5887	750	6	barasara	barasara	PROPN
ejpam-5887	750	7	s.	s.	PROPN
ejpam-5887	750	8	k.	k.	PROPN
ejpam-5887	750	9	vaidya	vaidya	PROPN
ejpam-5887	750	10	.	.	PUNCT
ejpam-5887	751	1	some	some	DET
ejpam-5887	751	2	new	new	ADJ
ejpam-5887	751	3	families	family	NOUN
ejpam-5887	751	4	of	of	ADP
ejpam-5887	751	5	edge	edge	NOUN
ejpam-5887	751	6	product	product	NOUN
ejpam-5887	751	7	cordial	cordial	ADJ
ejpam-5887	751	8	graphs	graph	NOUN
ejpam-5887	751	9	.	.	PUNCT
ejpam-5887	752	1	advanced	advanced	ADJ
ejpam-5887	752	2	modeling	modeling	NOUN
ejpam-5887	752	3	and	and	CCONJ
ejpam-5887	752	4	optimization	optimization	NOUN
ejpam-5887	752	5	,	,	PUNCT
ejpam-5887	752	6	15(1):103–111	15(1):103–111	PROPN
ejpam-5887	752	7	,	,	PUNCT
ejpam-5887	752	8	2013	2013	NUM
ejpam-5887	752	9	.	.	PUNCT
ejpam-5887	753	1	[	[	X
ejpam-5887	753	2	15	15	NUM
ejpam-5887	753	3	]	]	X
ejpam-5887	753	4	c.	c.	PROPN
ejpam-5887	753	5	m.	m.	PROPN
ejpam-5887	753	6	barasara	barasara	PROPN
ejpam-5887	753	7	s.	s.	PROPN
ejpam-5887	753	8	k.	k.	PROPN
ejpam-5887	753	9	vaidya	vaidya	PROPN
ejpam-5887	753	10	.	.	PROPN
ejpam-5887	754	1	on	on	ADP
ejpam-5887	754	2	edge	edge	NOUN
ejpam-5887	754	3	product	product	NOUN
ejpam-5887	754	4	cordial	cordial	ADJ
ejpam-5887	754	5	labeling	labeling	NOUN
ejpam-5887	754	6	of	of	ADP
ejpam-5887	754	7	some	some	DET
ejpam-5887	754	8	product	product	NOUN
ejpam-5887	754	9	related	relate	VERB
ejpam-5887	754	10	graphs	graph	NOUN
ejpam-5887	754	11	.	.	PUNCT
ejpam-5887	755	1	international	international	ADJ
ejpam-5887	755	2	journal	journal	NOUN
ejpam-5887	755	3	of	of	ADP
ejpam-5887	755	4	mathematics	mathematic	NOUN
ejpam-5887	755	5	and	and	CCONJ
ejpam-5887	755	6	its	its	PRON
ejpam-5887	755	7	applications	application	NOUN
ejpam-5887	755	8	,	,	PUNCT
ejpam-5887	755	9	2(2):15	2(2):15	NUM
ejpam-5887	755	10	–	–	PUNCT
ejpam-5887	755	11	22	22	NUM
ejpam-5887	755	12	,	,	PUNCT
ejpam-5887	755	13	2014	2014	NUM
ejpam-5887	755	14	.	.	PUNCT
ejpam-5887	756	1	[	[	X
ejpam-5887	756	2	16	16	NUM
ejpam-5887	756	3	]	]	X
ejpam-5887	756	4	c.	c.	PROPN
ejpam-5887	756	5	m.	m.	PROPN
ejpam-5887	756	6	barasara	barasara	PROPN
ejpam-5887	756	7	s.	s.	PROPN
ejpam-5887	756	8	k.	k.	PROPN
ejpam-5887	756	9	vaidya	vaidya	PROPN
ejpam-5887	756	10	.	.	PUNCT
ejpam-5887	756	11	product	product	NOUN
ejpam-5887	756	12	and	and	CCONJ
ejpam-5887	756	13	edge	edge	NOUN
ejpam-5887	756	14	product	product	NOUN
ejpam-5887	756	15	cordial	cordial	ADJ
ejpam-5887	756	16	labeling	labeling	NOUN
ejpam-5887	756	17	of	of	ADP
ejpam-5887	756	18	degree	degree	NOUN
ejpam-5887	756	19	splitting	splitting	NOUN
ejpam-5887	756	20	graph	graph	NOUN
ejpam-5887	756	21	of	of	ADP
ejpam-5887	756	22	some	some	DET
ejpam-5887	756	23	graphs	graph	NOUN
ejpam-5887	756	24	.	.	PUNCT
ejpam-5887	757	1	advances	advance	NOUN
ejpam-5887	757	2	and	and	CCONJ
ejpam-5887	757	3	applications	application	NOUN
ejpam-5887	757	4	in	in	ADP
ejpam-5887	757	5	discrete	discrete	ADJ
ejpam-5887	757	6	mathematics	mathematic	NOUN
ejpam-5887	757	7	,	,	PUNCT
ejpam-5887	757	8	15(1):61–74	15(1):61–74	NUM
ejpam-5887	757	9	,	,	PUNCT
ejpam-5887	757	10	2015	2015	NUM
ejpam-5887	757	11	.	.	PUNCT
ejpam-5887	758	1	[	[	X
ejpam-5887	758	2	17	17	NUM
ejpam-5887	758	3	]	]	X
ejpam-5887	758	4	n.	n.	PROPN
ejpam-5887	758	5	b.	b.	PROPN
ejpam-5887	758	6	patel	patel	PROPN
ejpam-5887	758	7	u.	u.	PROPN
ejpam-5887	758	8	m.	m.	PROPN
ejpam-5887	758	9	prajapati	prajapati	PROPN
ejpam-5887	758	10	.	.	PUNCT
ejpam-5887	759	1	edge	edge	NOUN
ejpam-5887	759	2	product	product	NOUN
ejpam-5887	759	3	cordial	cordial	ADJ
ejpam-5887	759	4	labeling	labeling	NOUN
ejpam-5887	759	5	of	of	ADP
ejpam-5887	759	6	some	some	DET
ejpam-5887	759	7	cycle	cycle	NOUN
ejpam-5887	759	8	related	relate	VERB
ejpam-5887	759	9	graphs	graph	NOUN
ejpam-5887	759	10	.	.	PUNCT
ejpam-5887	760	1	open	open	ADJ
ejpam-5887	760	2	journal	journal	NOUN
ejpam-5887	760	3	of	of	ADP
ejpam-5887	760	4	discrete	discrete	ADJ
ejpam-5887	760	5	mathematics	mathematic	NOUN
ejpam-5887	760	6	,	,	PUNCT
ejpam-5887	760	7	6:268–278	6:268–278	NUM
ejpam-5887	760	8	,	,	PUNCT
ejpam-5887	760	9	2016	2016	NUM
ejpam-5887	760	10	.	.	PUNCT
ejpam-5887	761	1	n.	n.	PROPN
ejpam-5887	761	2	m.	m.	PROPN
ejpam-5887	761	3	noureldeen	noureldeen	INTJ
ejpam-5887	761	4	et	et	PROPN
ejpam-5887	761	5	al	al	PROPN
ejpam-5887	761	6	.	.	PUNCT
ejpam-5887	761	7	/	/	SYM
ejpam-5887	761	8	eur	eur	PROPN
ejpam-5887	761	9	.	.	PUNCT
ejpam-5887	762	1	j.	j.	PROPN
ejpam-5887	762	2	pure	pure	PROPN
ejpam-5887	762	3	appl	appl	PROPN
ejpam-5887	762	4	.	.	PROPN
ejpam-5887	762	5	math	math	PROPN
ejpam-5887	762	6	,	,	PUNCT
ejpam-5887	762	7	18	18	NUM
ejpam-5887	762	8	(	(	PUNCT
ejpam-5887	762	9	2	2	NUM
ejpam-5887	762	10	)	)	PUNCT
ejpam-5887	762	11	(	(	PUNCT
ejpam-5887	762	12	2025	2025	NUM
ejpam-5887	762	13	)	)	PUNCT
ejpam-5887	762	14	,	,	PUNCT
ejpam-5887	762	15	5887	5887	NUM
ejpam-5887	762	16	21	21	NUM
ejpam-5887	762	17	of	of	ADP
ejpam-5887	762	18	21	21	NUM
ejpam-5887	762	19	[	[	SYM
ejpam-5887	762	20	18	18	NUM
ejpam-5887	762	21	]	]	X
ejpam-5887	762	22	n.	n.	PROPN
ejpam-5887	762	23	b.	b.	PROPN
ejpam-5887	762	24	patel	patel	PROPN
ejpam-5887	762	25	u.	u.	PROPN
ejpam-5887	762	26	m.	m.	PROPN
ejpam-5887	762	27	prajapati	prajapati	PROPN
ejpam-5887	762	28	.	.	PUNCT
ejpam-5887	763	1	edge	edge	NOUN
ejpam-5887	763	2	product	product	NOUN
ejpam-5887	763	3	cordial	cordial	ADJ
ejpam-5887	763	4	labeling	labeling	NOUN
ejpam-5887	763	5	of	of	ADP
ejpam-5887	763	6	some	some	DET
ejpam-5887	763	7	graphs	graph	NOUN
ejpam-5887	763	8	.	.	PUNCT
ejpam-5887	764	1	journal	journal	NOUN
ejpam-5887	764	2	of	of	ADP
ejpam-5887	764	3	applied	apply	VERB
ejpam-5887	764	4	mathematics	mathematic	NOUN
ejpam-5887	764	5	and	and	CCONJ
ejpam-5887	764	6	computational	computational	ADJ
ejpam-5887	764	7	mechanics	mechanic	NOUN
ejpam-5887	764	8	,	,	PUNCT
ejpam-5887	764	9	18(1):69–76	18(1):69–76	NUM
ejpam-5887	764	10	,	,	PUNCT
ejpam-5887	764	11	2019	2019	NUM
ejpam-5887	764	12	.	.	PUNCT
ejpam-5887	765	1	[	[	X
ejpam-5887	765	2	19	19	NUM
ejpam-5887	765	3	]	]	X
ejpam-5887	765	4	n.	n.	PROPN
ejpam-5887	765	5	b.	b.	PROPN
ejpam-5887	765	6	patel	patel	PROPN
ejpam-5887	765	7	u.	u.	PROPN
ejpam-5887	765	8	m.	m.	PROPN
ejpam-5887	765	9	prajapati	prajapati	PROPN
ejpam-5887	765	10	.	.	PUNCT
ejpam-5887	766	1	edge	edge	NOUN
ejpam-5887	766	2	product	product	NOUN
ejpam-5887	766	3	cordial	cordial	ADJ
ejpam-5887	766	4	labeling	labeling	NOUN
ejpam-5887	766	5	of	of	ADP
ejpam-5887	766	6	switching	switch	VERB
ejpam-5887	766	7	operations	operation	NOUN
ejpam-5887	766	8	on	on	ADP
ejpam-5887	766	9	some	some	DET
ejpam-5887	766	10	graphs	graph	NOUN
ejpam-5887	766	11	.	.	PUNCT
ejpam-5887	767	1	twms	twms	PROPN
ejpam-5887	767	2	j.	j.	PROPN
ejpam-5887	767	3	app	app	PROPN
ejpam-5887	767	4	.	.	PROPN
ejpam-5887	768	1	and	and	CCONJ
ejpam-5887	768	2	eng	eng	PROPN
ejpam-5887	768	3	.	.	PROPN
ejpam-5887	768	4	math	math	PROPN
ejpam-5887	768	5	.	.	PUNCT
ejpam-5887	769	1	,	,	PUNCT
ejpam-5887	769	2	12(1):191–199	12(1):191–199	PROPN
ejpam-5887	769	3	,	,	PUNCT
ejpam-5887	769	4	2022	2022	NUM
ejpam-5887	769	5	.	.	PUNCT
ejpam-5887	770	1	•	•	NUM
ejpam-5887	771	1	[	[	X
ejpam-5887	771	2	1–19	1–19	NOUN
ejpam-5887	771	3	]	]	PUNCT
ejpam-5887	771	4	are	be	AUX
ejpam-5887	771	5	journal	journal	ADJ
ejpam-5887	771	6	articles	article	NOUN
ejpam-5887	771	7	.	.	PUNCT